T(5,3)

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Knot presentations

 Planar diagram presentation X7,1,8,20 X14,2,15,1 X15,9,16,8 X2,10,3,9 X3,17,4,16 X10,18,11,17 X11,5,12,4 X18,6,19,5 X19,13,20,12 X6,14,7,13 Gauss code 2, -4, -5, 7, 8, -10, -1, 3, 4, -6, -7, 9, 10, -2, -3, 5, 6, -8, -9, 1 Dowker-Thistlethwaite code 14 -16 18 -20 2 -4 6 -8 10 -12

Polynomial invariants

 Alexander polynomial ${\displaystyle t^{4}-t^{3}+t-1+t^{-1}-t^{-3}+t^{-4}}$ Conway polynomial ${\displaystyle z^{8}+7z^{6}+14z^{4}+8z^{2}+1}$ 2nd Alexander ideal (db, data sources) ${\displaystyle \{1\}}$ Determinant and Signature { 1, 8 } Jones polynomial ${\displaystyle -q^{10}+q^{6}+q^{4}}$ HOMFLY-PT polynomial (db, data sources) ${\displaystyle z^{8}a^{-8}+8z^{6}a^{-8}-z^{6}a^{-10}+21z^{4}a^{-8}-7z^{4}a^{-10}+21z^{2}a^{-8}-14z^{2}a^{-10}+z^{2}a^{-12}+7a^{-8}-8a^{-10}+2a^{-12}}$ Kauffman polynomial (db, data sources) ${\displaystyle z^{8}a^{-8}+z^{8}a^{-10}+z^{7}a^{-9}+z^{7}a^{-11}-8z^{6}a^{-8}-8z^{6}a^{-10}-7z^{5}a^{-9}-7z^{5}a^{-11}+21z^{4}a^{-8}+21z^{4}a^{-10}+14z^{3}a^{-9}+14z^{3}a^{-11}-21z^{2}a^{-8}-22z^{2}a^{-10}-z^{2}a^{-12}-8za^{-9}-8za^{-11}+7a^{-8}+8a^{-10}+2a^{-12}}$ The A2 invariant Data:T(5,3)/QuantumInvariant/A2/1,0 The G2 invariant Data:T(5,3)/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {10_124,}

Same Jones Polynomial (up to mirroring, ${\displaystyle q\leftrightarrow q^{-1}}$): {10_124,}

Vassiliev invariants

 V2 and V3: (8, 20)
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 Data:T(5,3)/V 2,1 Data:T(5,3)/V 3,1 Data:T(5,3)/V 4,1 Data:T(5,3)/V 4,2 Data:T(5,3)/V 4,3 Data:T(5,3)/V 5,1 Data:T(5,3)/V 5,2 Data:T(5,3)/V 5,3 Data:T(5,3)/V 5,4 Data:T(5,3)/V 6,1 Data:T(5,3)/V 6,2 Data:T(5,3)/V 6,3 Data:T(5,3)/V 6,4 Data:T(5,3)/V 6,5 Data:T(5,3)/V 6,6 Data:T(5,3)/V 6,7 Data:T(5,3)/V 6,8 Data:T(5,3)/V 6,9

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=}$8 is the signature of T(5,3). Nonzero entries off the critical diagonals (if any exist) are highlighted in red.
 \ r \ j \
01234567χ
21       1-1
19     1  -1
17     11 0
15   11   0
13    1   1
11  1     1
91       1
71       1
Integral Khovanov Homology
 ${\displaystyle \dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}}$ ${\displaystyle i=5}$ ${\displaystyle i=7}$ ${\displaystyle i=9}$ ${\displaystyle r=0}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=1}$ ${\displaystyle r=2}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=3}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=4}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=5}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=6}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=7}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory. See A Sample KnotTheory Session.