# 9 41

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 Three-fold symmetric decorative knot Three-fold symmetric decorative knot in circle

### Knot presentations

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

Minimum Braid Representative A Morse Link Presentation An Arc Presentation

Length is 12, width is 5,

Braid index is 5

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

### Three dimensional invariants

 Symmetry type Reversible Unknotting number 2 3-genus 2 Bridge index 3 Super bridge index 4 Nakanishi index 2 Maximal Thurston-Bennequin number [-7][-4] Hyperbolic Volume 12.0989 A-Polynomial See Data:9 41/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 ${\displaystyle 3t^{2}-12t+19-12t^{-1}+3t^{-2}}$ Conway polynomial ${\displaystyle 3z^{4}+1}$ 2nd Alexander ideal (db, data sources) ${\displaystyle \{7,t+1\}}$ Determinant and Signature { 49, 0 } Jones polynomial ${\displaystyle -q^{3}+3q^{2}-5q+8-8q^{-1}+8q^{-2}-7q^{-3}+5q^{-4}-3q^{-5}+q^{-6}}$ HOMFLY-PT polynomial (db, data sources) ${\displaystyle a^{6}-3z^{2}a^{4}-3a^{4}+2z^{4}a^{2}+4z^{2}a^{2}+3a^{2}+z^{4}-z^{2}a^{-2}}$ Kauffman polynomial (db, data sources) ${\displaystyle 2a^{4}z^{8}+2a^{2}z^{8}+3a^{5}z^{7}+9a^{3}z^{7}+6az^{7}+a^{6}z^{6}-a^{4}z^{6}+5a^{2}z^{6}+7z^{6}-10a^{5}z^{5}-26a^{3}z^{5}-11az^{5}+5z^{5}a^{-1}-3a^{6}z^{4}-12a^{4}z^{4}-23a^{2}z^{4}+3z^{4}a^{-2}-11z^{4}+9a^{5}z^{3}+19a^{3}z^{3}+6az^{3}-3z^{3}a^{-1}+z^{3}a^{-3}+3a^{6}z^{2}+13a^{4}z^{2}+17a^{2}z^{2}-z^{2}a^{-2}+6z^{2}-2a^{5}z-4a^{3}z-2az-a^{6}-3a^{4}-3a^{2}}$ The A2 invariant ${\displaystyle q^{20}+q^{18}-2q^{16}-q^{12}-2q^{10}+2q^{8}+2q^{4}+q^{2}+2q^{-2}-2q^{-4}+q^{-6}+q^{-8}-q^{-10}}$ The G2 invariant ${\displaystyle q^{94}-2q^{92}+6q^{90}-10q^{88}+11q^{86}-7q^{84}-7q^{82}+27q^{80}-39q^{78}+44q^{76}-28q^{74}-3q^{72}+40q^{70}-66q^{68}+70q^{66}-45q^{64}+q^{62}+37q^{60}-64q^{58}+54q^{56}-25q^{54}-16q^{52}+42q^{50}-49q^{48}+27q^{46}+7q^{44}-43q^{42}+63q^{40}-60q^{38}+37q^{36}+6q^{34}-46q^{32}+79q^{30}-82q^{28}+64q^{26}-19q^{24}-28q^{22}+65q^{20}-77q^{18}+58q^{16}-17q^{14}-25q^{12}+51q^{10}-48q^{8}+18q^{6}+19q^{4}-46q^{2}+51-30q^{-2}-3q^{-4}+35q^{-6}-51q^{-8}+53q^{-10}-32q^{-12}+8q^{-14}+16q^{-16}-32q^{-18}+33q^{-20}-27q^{-22}+18q^{-24}-7q^{-26}-2q^{-28}+9q^{-30}-14q^{-32}+13q^{-34}-10q^{-36}+7q^{-38}-2q^{-40}-q^{-42}+2q^{-44}-4q^{-46}+3q^{-48}-2q^{-50}+q^{-52}}$

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

Same Alexander/Conway Polynomial: {K11n83,}

Same Jones Polynomial (up to mirroring, ${\displaystyle q\leftrightarrow q^{-1}}$): {K11n4, K11n21,}

### Vassiliev invariants

 V2 and V3: (0, 1)
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 ${\displaystyle 0}$ ${\displaystyle 8}$ ${\displaystyle 0}$ ${\displaystyle -48}$ ${\displaystyle -24}$ ${\displaystyle 0}$ ${\displaystyle {\frac {368}{3}}}$ ${\displaystyle -{\frac {64}{3}}}$ ${\displaystyle 104}$ ${\displaystyle 0}$ ${\displaystyle 32}$ ${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle -136}$ ${\displaystyle 296}$ ${\displaystyle -328}$ ${\displaystyle -88}$ ${\displaystyle -72}$

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 9 41. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.
 \ r \ j \
-6-5-4-3-2-10123χ
7         1-1
5        2 2
3       31 -2
1      52  3
-1     44   0
-3    44    0
-5   34     1
-7  24      -2
-9 13       2
-11 2        -2
-131         1
Integral Khovanov Homology
 ${\displaystyle \dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}}$ ${\displaystyle i=-1}$ ${\displaystyle i=1}$ ${\displaystyle r=-6}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-5}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-4}$ ${\displaystyle {\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=-3}$ ${\displaystyle {\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{3}}$ ${\displaystyle {\mathbb {Z} }^{3}}$ ${\displaystyle r=-2}$ ${\displaystyle {\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{4}}$ ${\displaystyle {\mathbb {Z} }^{4}}$ ${\displaystyle r=-1}$ ${\displaystyle {\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{4}}$ ${\displaystyle {\mathbb {Z} }^{4}}$ ${\displaystyle r=0}$ ${\displaystyle {\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{4}}$ ${\displaystyle {\mathbb {Z} }^{5}}$ ${\displaystyle r=1}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{3}}$ ${\displaystyle {\mathbb {Z} }^{3}}$ ${\displaystyle r=2}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=3}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$