In this section, we define the linking number for disoriented links and prove that the linking number of a disoriented link is its invariant.

#### Proof

Let *D* be a disoriented regular diagram of the link *L* with two components. We suppose that *D*′ is another regular diagram of *L*. From our discussions so far, we know that we may obtain *D*′ by performing, if necessary several times, the Reidemeister moves in Fig. 4, 5 and 6. Therefore, in order to prove the theorem, it is sufficient to show that the value of the linking number remains unchanged after each of the Reidemeister moves is performed on *D*.

*The move RI*: At the crossings of *D* at which we intend to apply the move *RI*, every section (edge) of such a crossing belongs to the same component. Therefore, applying the move *RI* does not affect the calculation of the linking number. In the same way, the move *RI*′ does also not affect the calculation of the linking number.

*The moves RII*: We shall only examine the effects of the cases
$\begin{array}{}{J}_{0}^{\prime},{J}_{2}^{\prime},{L}_{0}^{\prime}\text{\hspace{0.17em}and\hspace{0.17em}}{L}_{1}^{\prime}\end{array}$ of the moves *RII* in Fig. 5 to the calculation of linking number. The remaining cases of the moves *RII* can be examined in a similar way. An application of the moves *RII* on *D* only has an effect on the linking number if *A* and *B* belong to different components, see Fig. 7. In the cases (*a*) and (*c*), the crossing *c*_{1} is disoriented and the crossing *c*_{2} is oriented. Also, the crossings *c*_{1} and *c*_{2} have the same signs. By Definition 3.1, we get *ε*(*c*_{1}) − *ε*(*c*_{2}) = 0. In the cases (*b*) and (*d*), both crossings are disoriented. Since the crossings *c*_{1} and *c*_{2} have opposite signs, we get *ε*(*c*_{1})+*ε*(*c*_{2}) = 0. So, in each case, the linking number is unchanged under the second Reidemeister moves.

Fig. 7 Some Second Reidemeister Moves.

*The moves RIII*: Finally, let us consider the effect of the moves *RIII* on *D*. We only consider the effect of the cases *T*_{0} and *S*_{0},
$\begin{array}{}{S}_{0},{T}_{0}^{\prime}\text{\hspace{0.17em}and\hspace{0.17em}}{S}_{0}^{\prime},{T}_{1}^{\u2033}\text{\hspace{0.17em}and\hspace{0.17em}}{S}_{1}^{\u2033},\end{array}$ *T*_{2} and *S*_{2} of the moves *RIII* in Fig. 6 to the calculation of linking number. That is, we only examine the effect on the signs of the crossings *c*_{1}, *c*_{2}, *c*_{3} and
$\begin{array}{}{c}_{1}^{\prime},{c}_{2}^{\prime},{c}_{3}^{\prime}\end{array}$ in Fig. 8.

Fig. 8 Some Disoriented Third Reidemeister Moves.

To be identical the effects of the cases *T*_{i} and *S*_{i}, the following equations should always be hold:

If all crossings are oriented or disoriented,
$$\begin{array}{}\epsilon ({c}_{1})=\epsilon ({c}_{1}^{\prime}),\phantom{\rule{mediummathspace}{0ex}}\phantom{\rule{mediummathspace}{0ex}}\phantom{\rule{mediummathspace}{0ex}}\phantom{\rule{mediummathspace}{0ex}}\epsilon ({c}_{2})=\epsilon ({c}_{3}^{\prime}),\phantom{\rule{mediummathspace}{0ex}}\phantom{\rule{mediummathspace}{0ex}}\phantom{\rule{mediummathspace}{0ex}}\phantom{\rule{mediummathspace}{0ex}}\epsilon ({c}_{3})=\epsilon ({c}_{2}^{\prime}).\end{array}$$

If one of the crossings *c*_{2} and
$\begin{array}{}{c}_{3}^{\prime}\end{array}$ is oriented while other is disoriented or one of the crossings
*c*_{3} and
$\begin{array}{}{c}_{2}^{\prime}\end{array}$ is oriented while other is disoriented
$$\begin{array}{}\epsilon ({c}_{1})=\epsilon ({c}_{1}^{\prime}),\phantom{\rule{mediummathspace}{0ex}}\phantom{\rule{mediummathspace}{0ex}}\phantom{\rule{mediummathspace}{0ex}}\phantom{\rule{mediummathspace}{0ex}}\epsilon ({c}_{2})=-\epsilon ({c}_{3}^{\prime}),\phantom{\rule{mediummathspace}{0ex}}\phantom{\rule{mediummathspace}{0ex}}\phantom{\rule{mediummathspace}{0ex}}\phantom{\rule{mediummathspace}{0ex}}\epsilon ({c}_{3})=\epsilon ({c}_{2}^{\prime})\end{array}$$

or
$$\begin{array}{}\epsilon ({c}_{1})=\epsilon ({c}_{1}^{\prime}),\phantom{\rule{mediummathspace}{0ex}}\phantom{\rule{mediummathspace}{0ex}}\phantom{\rule{mediummathspace}{0ex}}\phantom{\rule{mediummathspace}{0ex}}\epsilon ({c}_{2})=\epsilon ({c}_{3}^{\prime}),\phantom{\rule{mediummathspace}{0ex}}\phantom{\rule{mediummathspace}{0ex}}\phantom{\rule{mediummathspace}{0ex}}\phantom{\rule{mediummathspace}{0ex}}\epsilon ({c}_{3})=-\epsilon ({c}_{2}^{\prime}).\end{array}$$

If one of the crossings *c*_{2} and
$\begin{array}{}{c}_{3}^{\prime}\end{array}$ is oriented while other is disoriented and one of the crossings *c*_{3} and
$\begin{array}{}{c}_{2}^{\prime}\end{array}$ is oriented while other is disoriented,
$$\begin{array}{}\epsilon ({c}_{1})=\epsilon ({c}_{1}^{\prime}),\phantom{\rule{mediummathspace}{0ex}}\phantom{\rule{mediummathspace}{0ex}}\phantom{\rule{mediummathspace}{0ex}}\phantom{\rule{mediummathspace}{0ex}}\epsilon ({c}_{2})=-\epsilon ({c}_{3}^{\prime}),\phantom{\rule{mediummathspace}{0ex}}\phantom{\rule{mediummathspace}{0ex}}\phantom{\rule{mediummathspace}{0ex}}\phantom{\rule{mediummathspace}{0ex}}\epsilon ({c}_{3})=-\epsilon ({c}_{2}^{\prime}).\end{array}$$

If *A*, *B* and *C* belong to the same component for the only case in Fig. 8, then the linking number is unaffected. So we suppose that *A* belong to a different component than *B* and *C*. Then the parts that have an effect on linking number is the sum of the signs of the crossings *c*_{1}, *c*_{2} and
$\begin{array}{}{c}_{1}^{\prime},{c}_{3}^{\prime}.\end{array}$ Since the crossings are oriented in the case (*a*), we have
$$\begin{array}{}\epsilon ({c}_{1})+\epsilon ({c}_{2})=\epsilon ({c}_{1}^{\prime})+\epsilon ({c}_{3}^{\prime})\end{array}$$

by Definition 3.1. Since the crossing
$\begin{array}{}{c}_{3}^{\prime}\end{array}$ is disoriented and others are oriented in the case (*b*), we have
$$\begin{array}{}\epsilon ({c}_{1})+\epsilon ({c}_{2})=\epsilon ({c}_{1}^{\prime})-\epsilon ({c}_{3}^{\prime}).\end{array}$$

Since the crossing
$\begin{array}{}{c}_{3}^{\prime}\end{array}$ is oriented and others are disoriented in the case (*c*), we have
$$\begin{array}{}\epsilon ({c}_{1})+\epsilon ({c}_{2})=\epsilon ({c}_{3}^{\prime})-\epsilon ({c}_{1}^{\prime}).\end{array}$$

Since the crossing *c*_{1} and
$\begin{array}{}{c}_{1}^{\prime}\end{array}$ are disoriented and others are oriented in the case (*d*), we have
$$\begin{array}{}\epsilon ({c}_{2})-\epsilon ({c}_{1})=\epsilon ({c}_{3}^{\prime})-\epsilon ({c}_{1}^{\prime}).\end{array}$$

Thus, none of the sums does cause any change to the linking number. The other cases, (i.e. the various possibilities for the components that *A*, *B* and *C* belong to) can be treated in a similar manner. The remaining cases of the moves *RIII* can be examined in a similar way. Hence, the linking number remains unchanged when we apply the moves *RIII*. □

We suppose now that *L* is a disoriented link with *n* components, *K*_{1}, *K*_{2}, …, *K*_{n} With regard to two components, *K*_{i} and *K*_{j}, *i* < *j*, we may define as an extension of the linking number *lk*(*L*) = *lk*(*K*_{i}, *K*_{j}), 1 ≤ *i* < *j* ≤ *n* This approach will give us *n*(*n* − 1)/2 linking numbers, and their sum,
$$\begin{array}{}{\displaystyle lk(L)=\sum _{1\le i<j\le n}lk({K}_{i},{K}_{j})}\end{array}$$

is called the total linking number of *K*. One can show that, in fact, the total linking number of *K* is an invariant of *K*.

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