# 10 161

 (KnotPlot image) See the full Rolfsen Knot Table. Visit 10 161's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 10 161 at Knotilus!

Warning. In 1973 K. Perko noticed that the knots that were later labeled 10161 and 10162 in Rolfsen's tables (which were published in 1976 and were based on earlier tables by Little (1900) and Conway (1970)) are in fact the same. In our table we removed Rolfsen's 10162 and renumbered the subsequent knots, so that our 10 crossings total is 165, one less than Rolfsen's 166. Read more: [1] [2] [3] [4].

### Knot presentations

 Planar diagram presentation X1425 X3,12,4,13 X7,14,8,15 X18,9,19,10 X6,19,7,20 X16,5,17,6 X10,17,11,18 X13,8,14,9 X20,15,1,16 X11,2,12,3 Gauss code -1, 10, -2, 1, 6, -5, -3, 8, 4, -7, -10, 2, -8, 3, 9, -6, 7, -4, 5, -9 Dowker-Thistlethwaite code 4 12 -16 14 -18 2 8 -20 -10 -6 Conway Notation [3:-20:-20]

Minimum Braid Representative A Morse Link Presentation An Arc Presentation

Length is 10, width is 3,

Braid index is 3

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

### Three dimensional invariants

 Symmetry type Reversible Unknotting number 3 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index 1 Maximal Thurston-Bennequin number [-14][5] Hyperbolic Volume 5.63877 A-Polynomial See Data:10 161/A-polynomial

### Four dimensional invariants

 Smooth 4 genus ${\displaystyle 3}$ Topological 4 genus ${\displaystyle 3}$ Concordance genus ${\displaystyle 3}$ Rasmussen s-Invariant -6

### Polynomial invariants

 Alexander polynomial ${\displaystyle t^{3}-2t+3-2t^{-1}+t^{-3}}$ Conway polynomial ${\displaystyle z^{6}+6z^{4}+7z^{2}+1}$ 2nd Alexander ideal (db, data sources) ${\displaystyle \{1\}}$ Determinant and Signature { 5, -4 } Jones polynomial ${\displaystyle q^{-3}+q^{-6}-q^{-7}+q^{-8}-q^{-9}+q^{-10}-q^{-11}}$ HOMFLY-PT polynomial (db, data sources) ${\displaystyle -z^{2}a^{10}-a^{10}-z^{2}a^{8}-a^{8}+z^{6}a^{6}+6z^{4}a^{6}+9z^{2}a^{6}+3a^{6}}$ Kauffman polynomial (db, data sources) ${\displaystyle z^{5}a^{13}-4z^{3}a^{13}+3za^{13}+z^{6}a^{12}-4z^{4}a^{12}+3z^{2}a^{12}+z^{5}a^{11}-3z^{3}a^{11}+za^{11}+z^{4}a^{10}-3z^{2}a^{10}+a^{10}-z^{4}a^{8}+3z^{2}a^{8}-a^{8}-z^{3}a^{7}+2za^{7}+z^{6}a^{6}-6z^{4}a^{6}+9z^{2}a^{6}-3a^{6}}$ The A2 invariant ${\displaystyle -q^{34}-q^{30}-q^{28}+q^{22}+q^{18}+q^{16}+q^{14}+q^{12}+q^{10}}$ The G2 invariant ${\displaystyle q^{176}+q^{172}-q^{170}+q^{166}-q^{164}+q^{162}+q^{160}-q^{158}+q^{156}-q^{154}-q^{152}+2q^{150}-4q^{148}-2q^{142}+q^{140}-2q^{138}-q^{136}-2q^{132}-q^{130}-q^{126}+q^{124}+q^{118}+q^{116}-q^{112}+2q^{110}-q^{108}+2q^{106}-2q^{102}+2q^{100}-q^{98}-q^{96}-2q^{92}+q^{88}-2q^{86}+2q^{84}+q^{78}-q^{76}+2q^{74}+2q^{72}+q^{70}+q^{68}+q^{66}+q^{64}+2q^{62}+q^{58}+q^{56}+q^{52}+q^{50}}$

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

Same Alexander/Conway Polynomial: {}

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

### Vassiliev invariants

 V2 and V3: (7, -18)
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 ${\displaystyle 28}$ ${\displaystyle -144}$ ${\displaystyle 392}$ ${\displaystyle {\frac {2882}{3}}}$ ${\displaystyle {\frac {430}{3}}}$ ${\displaystyle -4032}$ ${\displaystyle -7104}$ ${\displaystyle -1248}$ ${\displaystyle -912}$ ${\displaystyle {\frac {10976}{3}}}$ ${\displaystyle 10368}$ ${\displaystyle {\frac {80696}{3}}}$ ${\displaystyle {\frac {12040}{3}}}$ ${\displaystyle {\frac {1607257}{30}}}$ ${\displaystyle {\frac {25966}{15}}}$ ${\displaystyle {\frac {914594}{45}}}$ ${\displaystyle {\frac {6599}{18}}}$ ${\displaystyle {\frac {78457}{30}}}$

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 ${\displaystyle s=}$-4 is the signature of 10 161. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.
 \ r \ j \
-9-8-7-6-5-4-3-2-10χ
-5         11
-7         11
-9      11  0
-11     1    1
-13    121   0
-15   11     0
-17   11     0
-19 11       0
-21          0
-231         -1
Integral Khovanov Homology
 ${\displaystyle \dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}}$ ${\displaystyle i=-7}$ ${\displaystyle i=-5}$ ${\displaystyle i=-3}$ ${\displaystyle r=-9}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-8}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-7}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-6}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-5}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-4}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-3}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-2}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-1}$ ${\displaystyle r=0}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle {\mathbb {Z} }}$