# 10 132

 (KnotPlot image) See the full Rolfsen Knot Table. Visit 10 132's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 10 132 at Knotilus!

### Knot presentations

 Planar diagram presentation X4251 X8493 X5,12,6,13 X15,18,16,19 X9,16,10,17 X17,10,18,11 X13,20,14,1 X19,14,20,15 X11,6,12,7 X2837 Gauss code 1, -10, 2, -1, -3, 9, 10, -2, -5, 6, -9, 3, -7, 8, -4, 5, -6, 4, -8, 7 Dowker-Thistlethwaite code 4 8 -12 2 -16 -6 -20 -18 -10 -14 Conway Notation [23,3,2-]

Minimum Braid Representative A Morse Link Presentation An Arc Presentation

Length is 11, width is 4,

Braid index is 4

[{3, 10}, {2, 4}, {1, 3}, {13, 11}, {10, 12}, {11, 8}, {7, 9}, {8, 5}, {4, 6}, {5, 7}, {6, 13}, {12, 2}, {9, 1}]

### Three dimensional invariants

 Symmetry type Reversible Unknotting number 1 3-genus 2 Bridge index 3 Super bridge index Missing Nakanishi index 1 Maximal Thurston-Bennequin number [-8][-1] Hyperbolic Volume 4.05686 A-Polynomial See Data:10 132/A-polynomial

[edit Notes for 10 132's three dimensional invariants] 10 132 is a very interesting knot from the point of view of contact geometry. In particular, it is a transversely nonsimple knot, and it was the last knot with at most 10 crossings for which the maximal Thurston-Bennequin number was calculated.

### Four dimensional invariants

 Smooth 4 genus ${\displaystyle 1}$ Topological 4 genus ${\displaystyle 1}$ Concordance genus ${\displaystyle 2}$ Rasmussen s-Invariant 2

### Polynomial invariants

 Alexander polynomial ${\displaystyle t^{2}-t+1-t^{-1}+t^{-2}}$ Conway polynomial ${\displaystyle z^{4}+3z^{2}+1}$ 2nd Alexander ideal (db, data sources) ${\displaystyle \{1\}}$ Determinant and Signature { 5, 0 } Jones polynomial ${\displaystyle q^{-2}+q^{-4}-q^{-5}+q^{-6}-q^{-7}}$ HOMFLY-PT polynomial (db, data sources) ${\displaystyle -z^{2}a^{6}-2a^{6}+z^{4}a^{4}+4z^{2}a^{4}+3a^{4}}$ Kauffman polynomial (db, data sources) ${\displaystyle a^{6}z^{8}+a^{4}z^{8}+a^{7}z^{7}+2a^{5}z^{7}+a^{3}z^{7}-6a^{6}z^{6}-6a^{4}z^{6}-6a^{7}z^{5}-12a^{5}z^{5}-6a^{3}z^{5}+10a^{6}z^{4}+10a^{4}z^{4}+10a^{7}z^{3}+19a^{5}z^{3}+9a^{3}z^{3}-6a^{6}z^{2}-7a^{4}z^{2}-a^{2}z^{2}-5a^{7}z-8a^{5}z-4a^{3}z-az+2a^{6}+3a^{4}}$ The A2 invariant ${\displaystyle -q^{22}-q^{20}-q^{18}+q^{14}+q^{12}+2q^{10}+q^{8}+q^{6}}$ The G2 invariant ${\displaystyle q^{108}+q^{104}-q^{100}-q^{92}-q^{90}-q^{86}-q^{84}-q^{82}-2q^{80}-q^{78}-q^{76}-2q^{74}-q^{68}-q^{64}+q^{62}+q^{60}+q^{58}+q^{56}+q^{54}+2q^{52}+3q^{50}+q^{48}+q^{46}+2q^{44}+q^{42}+2q^{40}+q^{38}+q^{34}+q^{32}-q^{28}+q^{26}-q^{24}-q^{18}+q^{16}-q^{12}+q^{4}-q^{2}+1+q^{-6}-q^{-8}}$

### "Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {5_1,}

Same Jones Polynomial (up to mirroring, ${\displaystyle q\leftrightarrow q^{-1}}$): {5_1,}

### Vassiliev invariants

 V2 and V3: (3, -5)
V2,1 through V6,9:
 V2,1 V3,1 V4,1 V4,2 V4,3 V5,1 V5,2 V5,3 V5,4 V6,1 V6,2 V6,3 V6,4 V6,5 V6,6 V6,7 V6,8 V6,9 ${\displaystyle 12}$ ${\displaystyle -40}$ ${\displaystyle 72}$ ${\displaystyle 174}$ ${\displaystyle 26}$ ${\displaystyle -480}$ ${\displaystyle -{\frac {2608}{3}}}$ ${\displaystyle -{\frac {352}{3}}}$ ${\displaystyle -168}$ ${\displaystyle 288}$ ${\displaystyle 800}$ ${\displaystyle 2088}$ ${\displaystyle 312}$ ${\displaystyle {\frac {42751}{10}}}$ ${\displaystyle -{\frac {3506}{15}}}$ ${\displaystyle {\frac {10034}{5}}}$ ${\displaystyle {\frac {203}{2}}}$ ${\displaystyle {\frac {2751}{10}}}$

V2,1 through V6,9 were provided by Petr Dunin-Barkowski <barkovs@itep.ru>, Andrey Smirnov <asmirnov@itep.ru>, and Alexei Sleptsov <sleptsov@itep.ru> and uploaded on October 2010 by User:Drorbn. Note that they are normalized differently than V2 and V3.

### Khovanov Homology

The coefficients of the monomials ${\displaystyle t^{r}q^{j}}$ are shown, along with their alternating sums ${\displaystyle \chi }$ (fixed ${\displaystyle j}$, alternation over ${\displaystyle r}$). The squares with yellow highlighting are those on the "critical diagonals", where ${\displaystyle j-2r=s+1}$ or ${\displaystyle j-2r=s-1}$, where ${\displaystyle s=}$0 is the signature of 10 132. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.
 \ r \ j \
-7-6-5-4-3-2-10χ
-1      110
-3       11
-5    12  1
-7   1    1
-9   11   0
-11 11     0
-13        0
-151       -1
Integral Khovanov Homology
 ${\displaystyle \dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}}$ ${\displaystyle i=-3}$ ${\displaystyle i=-1}$ ${\displaystyle i=1}$ ${\displaystyle r=-7}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-6}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-5}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-4}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-3}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-2}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=-1}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=0}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle {\mathbb {Z} }}$