# 6 3

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 3D depiction Irish knot, sum of four 6.3

### Knot presentations

 Planar diagram presentation X4251 X8493 X12,9,1,10 X10,5,11,6 X6,11,7,12 X2837 Gauss code 1, -6, 2, -1, 4, -5, 6, -2, 3, -4, 5, -3 Dowker-Thistlethwaite code 4 8 10 2 12 6 Conway Notation [2112]

Minimum Braid Representative A Morse Link Presentation An Arc Presentation

Length is 6, width is 3,

Braid index is 3

[{3, 7}, {2, 5}, {4, 6}, {5, 8}, {7, 9}, {8, 4}, {1, 3}, {9, 2}, {6, 1}]
 Knot 6_3. A graph, knot 6_3.

### Three dimensional invariants

 Symmetry type Fully amphicheiral Unknotting number 1 3-genus 2 Bridge index 2 Super bridge index ${\displaystyle \{3,4\}}$ Nakanishi index 1 Maximal Thurston-Bennequin number [-4][-4] Hyperbolic Volume 5.69302 A-Polynomial See Data:6 3/A-polynomial

### Four dimensional invariants

 Smooth 4 genus ${\displaystyle 1}$ Topological 4 genus ${\displaystyle 1}$ Concordance genus ${\displaystyle 2}$ Rasmussen s-Invariant 0

### Polynomial invariants

 Alexander polynomial ${\displaystyle t^{2}-3t+5-3t^{-1}+t^{-2}}$ Conway polynomial ${\displaystyle z^{4}+z^{2}+1}$ 2nd Alexander ideal (db, data sources) ${\displaystyle \{1\}}$ Determinant and Signature { 13, 0 } Jones polynomial ${\displaystyle -q^{3}+2q^{2}-2q+3-2q^{-1}+2q^{-2}-q^{-3}}$ HOMFLY-PT polynomial (db, data sources) ${\displaystyle z^{4}-a^{2}z^{2}-z^{2}a^{-2}+3z^{2}-a^{2}-a^{-2}+3}$ Kauffman polynomial (db, data sources) ${\displaystyle az^{5}+z^{5}a^{-1}+2a^{2}z^{4}+2z^{4}a^{-2}+4z^{4}+a^{3}z^{3}+az^{3}+z^{3}a^{-1}+z^{3}a^{-3}-3a^{2}z^{2}-3z^{2}a^{-2}-6z^{2}-a^{3}z-2az-2za^{-1}-za^{-3}+a^{2}+a^{-2}+3}$ The A2 invariant ${\displaystyle -q^{10}+2q^{2}+1+2q^{-2}-q^{-10}}$ The G2 invariant ${\displaystyle q^{52}-q^{50}+2q^{48}-2q^{46}-q^{44}+q^{42}-3q^{40}+4q^{38}-4q^{36}+q^{34}-3q^{30}+3q^{28}-3q^{26}+q^{24}+q^{22}-2q^{20}+q^{18}+q^{16}-q^{14}+4q^{12}-3q^{10}+3q^{8}+q^{6}-q^{4}+6q^{2}-5+6q^{-2}-q^{-4}+q^{-6}+3q^{-8}-3q^{-10}+4q^{-12}-q^{-14}+q^{-16}+q^{-18}-2q^{-20}+q^{-22}+q^{-24}-3q^{-26}+3q^{-28}-3q^{-30}+q^{-34}-4q^{-36}+4q^{-38}-3q^{-40}+q^{-42}-q^{-44}-2q^{-46}+2q^{-48}-q^{-50}+q^{-52}}$

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

Same Alexander/Conway Polynomial: {K11n12,}

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

### Vassiliev invariants

 V2 and V3: (1, 0)
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 ${\displaystyle 4}$ ${\displaystyle 0}$ ${\displaystyle 8}$ ${\displaystyle {\frac {14}{3}}}$ ${\displaystyle -{\frac {14}{3}}}$ ${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle {\frac {32}{3}}}$ ${\displaystyle 0}$ ${\displaystyle {\frac {56}{3}}}$ ${\displaystyle -{\frac {56}{3}}}$ ${\displaystyle {\frac {511}{30}}}$ ${\displaystyle {\frac {418}{15}}}$ ${\displaystyle -{\frac {1858}{45}}}$ ${\displaystyle {\frac {65}{18}}}$ ${\displaystyle -{\frac {449}{30}}}$

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=}$0 is the signature of 6 3. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.
 \ r \ j \
-3-2-10123χ
7      1-1
5     1 1
3    11 0
1   21  1
-1  12   1
-3 11    0
-5 1     1
-71      -1
Integral Khovanov Homology
 ${\displaystyle \dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}}$ ${\displaystyle i=-1}$ ${\displaystyle i=1}$ ${\displaystyle r=-3}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-2}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-1}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=0}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=1}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=2}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=3}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$