# 10 1

 (KnotPlot image) See the full Rolfsen Knot Table. Visit 10 1's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 10 1 at Knotilus!

### Knot presentations

 Planar diagram presentation X1425 X11,14,12,15 X3,13,4,12 X13,3,14,2 X5,20,6,1 X7,18,8,19 X9,16,10,17 X15,10,16,11 X17,8,18,9 X19,6,20,7 Gauss code -1, 4, -3, 1, -5, 10, -6, 9, -7, 8, -2, 3, -4, 2, -8, 7, -9, 6, -10, 5 Dowker-Thistlethwaite code 4 12 20 18 16 14 2 10 8 6 Conway Notation [82]

Minimum Braid Representative A Morse Link Presentation An Arc Presentation

Length is 13, width is 6,

Braid index is 6

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

### Three dimensional invariants

 Symmetry type Reversible Unknotting number 1 3-genus 1 Bridge index 2 Super bridge index Missing Nakanishi index 1 Maximal Thurston-Bennequin number [-9][-3] Hyperbolic Volume 3.5262 A-Polynomial See Data:10 1/A-polynomial

### Four dimensional invariants

 Smooth 4 genus ${\displaystyle 1}$ Topological 4 genus ${\displaystyle 1}$ Concordance genus ${\displaystyle 1}$ Rasmussen s-Invariant 0

### Polynomial invariants

 Alexander polynomial ${\displaystyle -4t+9-4t^{-1}}$ Conway polynomial ${\displaystyle 1-4z^{2}}$ 2nd Alexander ideal (db, data sources) ${\displaystyle \{1\}}$ Determinant and Signature { 17, 0 } Jones polynomial ${\displaystyle q^{2}-q+2-2q^{-1}+2q^{-2}-2q^{-3}+2q^{-4}-2q^{-5}+q^{-6}-q^{-7}+q^{-8}}$ HOMFLY-PT polynomial (db, data sources) ${\displaystyle a^{8}-z^{2}a^{6}-a^{6}-z^{2}a^{4}-z^{2}a^{2}-z^{2}+a^{-2}}$ Kauffman polynomial (db, data sources) ${\displaystyle a^{7}z^{9}+a^{5}z^{9}+a^{8}z^{8}+2a^{6}z^{8}+a^{4}z^{8}-7a^{7}z^{7}-6a^{5}z^{7}+a^{3}z^{7}-7a^{8}z^{6}-12a^{6}z^{6}-4a^{4}z^{6}+a^{2}z^{6}+16a^{7}z^{5}+12a^{5}z^{5}-3a^{3}z^{5}+az^{5}+15a^{8}z^{4}+21a^{6}z^{4}+3a^{4}z^{4}-2a^{2}z^{4}+z^{4}-14a^{7}z^{3}-11a^{5}z^{3}+a^{3}z^{3}-az^{3}+z^{3}a^{-1}-10a^{8}z^{2}-11a^{6}z^{2}+z^{2}a^{-2}+4a^{7}z+4a^{5}z+a^{8}+a^{6}-a^{-2}}$ The A2 invariant ${\displaystyle q^{26}+q^{24}-q^{18}-q^{16}+q^{-2}+q^{-6}+q^{-8}}$ The G2 invariant Data:10 1/QuantumInvariant/G2/1,0

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

Same Alexander/Conway Polynomial: {8_3,}

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

### Vassiliev invariants

 V2 and V3: (-4, 6)
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 ${\displaystyle -16}$ ${\displaystyle 48}$ ${\displaystyle 128}$ ${\displaystyle {\frac {232}{3}}}$ ${\displaystyle {\frac {200}{3}}}$ ${\displaystyle -768}$ ${\displaystyle -1088}$ ${\displaystyle -256}$ ${\displaystyle -272}$ ${\displaystyle -{\frac {2048}{3}}}$ ${\displaystyle 1152}$ ${\displaystyle -{\frac {3712}{3}}}$ ${\displaystyle -{\frac {3200}{3}}}$ ${\displaystyle {\frac {30898}{15}}}$ ${\displaystyle {\frac {5768}{15}}}$ ${\displaystyle -{\frac {6248}{45}}}$ ${\displaystyle {\frac {4046}{9}}}$ ${\displaystyle -{\frac {4142}{15}}}$

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 1. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.
 \ r \ j \
-8-7-6-5-4-3-2-1012χ
5          11
3           0
1        21 1
-1       11  0
-3      11   0
-5     11    0
-7    11     0
-9   11      0
-11   1       -1
-13 11        0
-15           0
-171          1
Integral Khovanov Homology
 ${\displaystyle \dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}}$ ${\displaystyle i=-1}$ ${\displaystyle i=1}$ ${\displaystyle r=-8}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-7}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-6}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-5}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-4}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-3}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-2}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-1}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=0}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=1}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=2}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$