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(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a315 at Knotilus!

Knot presentations

Planar diagram presentation X6271 X12,4,13,3 X16,5,17,6 X18,8,19,7 X22,10,1,9 X4,12,5,11 X20,13,21,14 X10,15,11,16 X2,17,3,18 X8,20,9,19 X14,21,15,22
Gauss code 1, -9, 2, -6, 3, -1, 4, -10, 5, -8, 6, -2, 7, -11, 8, -3, 9, -4, 10, -7, 11, -5
Dowker-Thistlethwaite code 6 12 16 18 22 4 20 10 2 8 14
A Braid Representative
A Morse Link Presentation K11a315 ML.gif

Three dimensional invariants

Symmetry type Chiral
Unknotting number \{1,2\}
3-genus 4
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a315/ThurstonBennequinNumber
Hyperbolic Volume 18.2725
A-Polynomial See Data:K11a315/A-polynomial

[edit Notes for K11a315's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 4
Rasmussen s-Invariant 0

[edit Notes for K11a315's four dimensional invariants]

Polynomial invariants

Alexander polynomial t^4-6 t^3+18 t^2-33 t+41-33 t^{-1} +18 t^{-2} -6 t^{-3} + t^{-4}
Conway polynomial z^8+2 z^6+2 z^4+z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 157, 0 }
Jones polynomial q^6-4 q^5+9 q^4-16 q^3+22 q^2-25 q+26-22 q^{-1} +17 q^{-2} -10 q^{-3} +4 q^{-4} - q^{-5}
HOMFLY-PT polynomial (db, data sources) z^8-a^2 z^6-2 z^6 a^{-2} +5 z^6-3 a^2 z^4-7 z^4 a^{-2} +z^4 a^{-4} +11 z^4-4 a^2 z^2-9 z^2 a^{-2} +2 z^2 a^{-4} +12 z^2-2 a^2-4 a^{-2} + a^{-4} +6
Kauffman polynomial (db, data sources) 3 z^{10} a^{-2} +3 z^{10}+10 a z^9+17 z^9 a^{-1} +7 z^9 a^{-3} +13 a^2 z^8+12 z^8 a^{-2} +7 z^8 a^{-4} +18 z^8+9 a^3 z^7-11 a z^7-32 z^7 a^{-1} -8 z^7 a^{-3} +4 z^7 a^{-5} +4 a^4 z^6-25 a^2 z^6-41 z^6 a^{-2} -14 z^6 a^{-4} +z^6 a^{-6} -55 z^6+a^5 z^5-13 a^3 z^5-2 a z^5+15 z^5 a^{-1} -6 z^5 a^{-3} -9 z^5 a^{-5} -4 a^4 z^4+23 a^2 z^4+39 z^4 a^{-2} +8 z^4 a^{-4} -2 z^4 a^{-6} +56 z^4-a^5 z^3+7 a^3 z^3+9 a z^3+3 z^3 a^{-1} +8 z^3 a^{-3} +6 z^3 a^{-5} -11 a^2 z^2-19 z^2 a^{-2} -3 z^2 a^{-4} +z^2 a^{-6} -26 z^2-2 a^3 z-4 a z-4 z a^{-1} -3 z a^{-3} -z a^{-5} +2 a^2+4 a^{-2} + a^{-4} +6
The A2 invariant -q^{14}+2 q^{12}-4 q^{10}+2 q^8+q^6-3 q^4+7 q^2-3+5 q^{-2} - q^{-4} -2 q^{-6} +3 q^{-8} -5 q^{-10} +2 q^{-12} - q^{-16} + q^{-18}
The G2 invariant Data:K11a315/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a24, K11a26,}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {K11a24, K11a26,}

Vassiliev invariants

V2 and V3: (1, -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
4 -8 8 -\frac{34}{3} -\frac{38}{3} -32 -\frac{176}{3} \frac{160}{3} -72 \frac{32}{3} 32 -\frac{136}{3} -\frac{152}{3} -\frac{209}{30} \frac{1738}{15} -\frac{3898}{45} -\frac{1039}{18} -\frac{1169}{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 t^rq^j are shown, along with their alternating sums \chi (fixed j, alternation over r). The squares with yellow highlighting are those on the "critical diagonals", where j-2r=s+1 or j-2r=s-1, where s=0 is the signature of K11a315. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
13           11
11          3 -3
9         61 5
7        103  -7
5       126   6
3      1310    -3
1     1312     1
-1    1014      4
-3   712       -5
-5  310        7
-7 17         -6
-9 3          3
-111           -1
Integral Khovanov Homology

(db, data source)

\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-1 i=1
r=-5 {\mathbb Z}
r=-4 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-3 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-2 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=-1 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=0 {\mathbb Z}^{14}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{13}
r=1 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{13} {\mathbb Z}^{13}
r=2 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{12}
r=3 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=4 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=5 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=6 {\mathbb Z}_2 {\mathbb Z}

Computer Talk

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

Modifying This Page

Read me first: Modifying Knot Pages.

See/edit the Hoste-Thistlethwaite Knot Page master template (intermediate).

See/edit the Hoste-Thistlethwaite_Splice_Base (expert).

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