# 10 22

 (KnotPlot image) See the full Rolfsen Knot Table. Visit 10 22's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 10 22 at Knotilus!

### Knot presentations

 Planar diagram presentation X6271 X16,12,17,11 X12,3,13,4 X2,15,3,16 X14,5,15,6 X18,8,19,7 X20,10,1,9 X8,20,9,19 X4,13,5,14 X10,18,11,17 Gauss code 1, -4, 3, -9, 5, -1, 6, -8, 7, -10, 2, -3, 9, -5, 4, -2, 10, -6, 8, -7 Dowker-Thistlethwaite code 6 12 14 18 20 16 4 2 10 8 Conway Notation [3313]

Minimum Braid Representative A Morse Link Presentation An Arc Presentation

Length is 11, width is 4,

Braid index is 4

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

### Three dimensional invariants

 Symmetry type Reversible Unknotting number 2 3-genus 3 Bridge index 2 Super bridge index Missing Nakanishi index 1 Maximal Thurston-Bennequin number [-5][-7] Hyperbolic Volume 9.98187 A-Polynomial See Data:10 22/A-polynomial

### Four dimensional invariants

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

### Polynomial invariants

 Alexander polynomial ${\displaystyle -2t^{3}+6t^{2}-10t+13-10t^{-1}+6t^{-2}-2t^{-3}}$ Conway polynomial ${\displaystyle -2z^{6}-6z^{4}-4z^{2}+1}$ 2nd Alexander ideal (db, data sources) ${\displaystyle \{1\}}$ Determinant and Signature { 49, 0 } Jones polynomial ${\displaystyle q^{6}-2q^{5}+4q^{4}-6q^{3}+7q^{2}-8q+8-6q^{-1}+4q^{-2}-2q^{-3}+q^{-4}}$ HOMFLY-PT polynomial (db, data sources) ${\displaystyle -z^{6}a^{-2}-z^{6}+a^{2}z^{4}-4z^{4}a^{-2}+z^{4}a^{-4}-4z^{4}+3a^{2}z^{2}-5z^{2}a^{-2}+3z^{2}a^{-4}-5z^{2}+2a^{2}-2a^{-2}+2a^{-4}-1}$ Kauffman polynomial (db, data sources) ${\displaystyle z^{9}a^{-1}+z^{9}a^{-3}+4z^{8}a^{-2}+2z^{8}a^{-4}+2z^{8}+3az^{7}-z^{7}a^{-3}+2z^{7}a^{-5}+3a^{2}z^{6}-12z^{6}a^{-2}-6z^{6}a^{-4}+z^{6}a^{-6}-2z^{6}+2a^{3}z^{5}-6az^{5}-z^{5}a^{-1}-7z^{5}a^{-5}+a^{4}z^{4}-6a^{2}z^{4}+16z^{4}a^{-2}+6z^{4}a^{-4}-4z^{4}a^{-6}-z^{4}-3a^{3}z^{3}+7az^{3}-4z^{3}a^{-3}+6z^{3}a^{-5}-2a^{4}z^{2}+6a^{2}z^{2}-12z^{2}a^{-2}-6z^{2}a^{-4}+4z^{2}a^{-6}+6z^{2}-az+za^{-1}+za^{-3}-za^{-5}-2a^{2}+2a^{-2}+2a^{-4}-1}$ The A2 invariant ${\displaystyle q^{12}+q^{8}+q^{6}-q^{4}+2q^{2}-1-q^{-4}-2q^{-6}+q^{-8}-q^{-10}+q^{-12}+q^{-14}+q^{-18}}$ The G2 invariant ${\displaystyle q^{66}-q^{64}+2q^{62}-3q^{60}+2q^{58}-q^{56}-2q^{54}+6q^{52}-7q^{50}+10q^{48}-10q^{46}+6q^{44}+q^{42}-10q^{40}+18q^{38}-22q^{36}+21q^{34}-15q^{32}+4q^{30}+12q^{28}-22q^{26}+33q^{24}-29q^{22}+21q^{20}-6q^{18}-10q^{16}+24q^{14}-27q^{12}+23q^{10}-3q^{8}-11q^{6}+19q^{4}-18q^{2}+3+17q^{-2}-36q^{-4}+35q^{-6}-27q^{-8}+29q^{-12}-52q^{-14}+54q^{-16}-43q^{-18}+14q^{-20}+13q^{-22}-39q^{-24}+48q^{-26}-42q^{-28}+24q^{-30}+q^{-32}-21q^{-34}+31q^{-36}-23q^{-38}+7q^{-40}+12q^{-42}-25q^{-44}+26q^{-46}-14q^{-48}-7q^{-50}+31q^{-52}-40q^{-54}+39q^{-56}-20q^{-58}-5q^{-60}+26q^{-62}-36q^{-64}+37q^{-66}-25q^{-68}+8q^{-70}+8q^{-72}-18q^{-74}+20q^{-76}-15q^{-78}+9q^{-80}-2q^{-82}-3q^{-84}+4q^{-86}-5q^{-88}+3q^{-90}-q^{-92}+q^{-94}}$

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

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, ${\displaystyle q\leftrightarrow q^{-1}}$): {10_35,}

### Vassiliev invariants

 V2 and V3: (-4, -2)
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 ${\displaystyle -16}$ ${\displaystyle -16}$ ${\displaystyle 128}$ ${\displaystyle {\frac {664}{3}}}$ ${\displaystyle {\frac {344}{3}}}$ ${\displaystyle 256}$ ${\displaystyle {\frac {1472}{3}}}$ ${\displaystyle {\frac {320}{3}}}$ ${\displaystyle 144}$ ${\displaystyle -{\frac {2048}{3}}}$ ${\displaystyle 128}$ ${\displaystyle -{\frac {10624}{3}}}$ ${\displaystyle -{\frac {5504}{3}}}$ ${\displaystyle -{\frac {44342}{15}}}$ ${\displaystyle {\frac {20408}{15}}}$ ${\displaystyle -{\frac {179048}{45}}}$ ${\displaystyle {\frac {5126}{9}}}$ ${\displaystyle -{\frac {15302}{15}}}$

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 22. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.
 \ r \ j \
-4-3-2-10123456χ
13          11
11         1 -1
9        31 2
7       31  -2
5      43   1
3     43    -1
1    44     0
-1   35      2
-3  13       -2
-5 13        2
-7 1         -1
-91          1
Integral Khovanov Homology
 ${\displaystyle \dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}}$ ${\displaystyle i=-1}$ ${\displaystyle i=1}$ ${\displaystyle r=-4}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-3}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-2}$ ${\displaystyle {\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-1}$ ${\displaystyle {\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{3}}$ ${\displaystyle {\mathbb {Z} }^{3}}$ ${\displaystyle r=0}$ ${\displaystyle {\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}}$ ${\displaystyle {\mathbb {Z} }^{4}}$ ${\displaystyle r=1}$ ${\displaystyle {\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{4}}$ ${\displaystyle {\mathbb {Z} }^{4}}$ ${\displaystyle r=2}$ ${\displaystyle {\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{4}}$ ${\displaystyle {\mathbb {Z} }^{4}}$ ${\displaystyle r=3}$ ${\displaystyle {\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{3}}$ ${\displaystyle {\mathbb {Z} }^{3}}$ ${\displaystyle r=4}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}}$ ${\displaystyle {\mathbb {Z} }^{3}}$ ${\displaystyle r=5}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=6}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$