K11a312

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K11a311.gif

K11a311

K11a313.gif

K11a313

Contents

K11a312.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a312 at Knotilus!



Knot presentations

Planar diagram presentation X6271 X12,3,13,4 X16,5,17,6 X14,8,15,7 X18,9,19,10 X20,11,21,12 X2,13,3,14 X22,16,1,15 X4,17,5,18 X10,19,11,20 X8,21,9,22
Gauss code 1, -7, 2, -9, 3, -1, 4, -11, 5, -10, 6, -2, 7, -4, 8, -3, 9, -5, 10, -6, 11, -8
Dowker-Thistlethwaite code 6 12 16 14 18 20 2 22 4 10 8
A Braid Representative
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A Morse Link Presentation K11a312 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number \{1,2\}
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a312/ThurstonBennequinNumber
Hyperbolic Volume 16.0409
A-Polynomial See Data:K11a312/A-polynomial

[edit Notes for K11a312's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 3
Rasmussen s-Invariant 2

[edit Notes for K11a312's four dimensional invariants]

Polynomial invariants

Alexander polynomial 3 t^3-13 t^2+27 t-33+27 t^{-1} -13 t^{-2} +3 t^{-3}
Conway polynomial 3 z^6+5 z^4+2 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 119, -2 }
Jones polynomial -q^2+3 q-6+12 q^{-1} -16 q^{-2} +19 q^{-3} -19 q^{-4} +17 q^{-5} -13 q^{-6} +8 q^{-7} -4 q^{-8} + q^{-9}
HOMFLY-PT polynomial (db, data sources) z^2 a^8+a^8-3 z^4 a^6-7 z^2 a^6-4 a^6+2 z^6 a^4+7 z^4 a^4+9 z^2 a^4+4 a^4+z^6 a^2+2 z^4 a^2+z^2 a^2-z^4-2 z^2
Kauffman polynomial (db, data sources) z^6 a^{10}-2 z^4 a^{10}+z^2 a^{10}+4 z^7 a^9-10 z^5 a^9+6 z^3 a^9+6 z^8 a^8-13 z^6 a^8+5 z^4 a^8-z^2 a^8+a^8+5 z^9 a^7-6 z^7 a^7-5 z^5 a^7+2 z^3 a^7+2 z^{10} a^6+7 z^8 a^6-27 z^6 a^6+29 z^4 a^6-18 z^2 a^6+4 a^6+11 z^9 a^5-28 z^7 a^5+31 z^5 a^5-16 z^3 a^5+2 z a^5+2 z^{10} a^4+8 z^8 a^4-33 z^6 a^4+48 z^4 a^4-25 z^2 a^4+4 a^4+6 z^9 a^3-13 z^7 a^3+16 z^5 a^3-6 z^3 a^3+2 z a^3+7 z^8 a^2-17 z^6 a^2+20 z^4 a^2-7 z^2 a^2+5 z^7 a-9 z^5 a+4 z^3 a+3 z^6-6 z^4+2 z^2+z^5 a^{-1} -2 z^3 a^{-1}
The A2 invariant Data:K11a312/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a312/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

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

Vassiliev invariants

V2 and V3: (2, -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
8 -16 32 \frac{28}{3} -\frac{76}{3} -128 \frac{224}{3} -\frac{160}{3} 208 \frac{256}{3} 128 \frac{224}{3} -\frac{608}{3} -\frac{7649}{15} \frac{6332}{5} -\frac{78356}{45} -\frac{2047}{9} -\frac{5249}{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 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=-2 is the signature of K11a312. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-8-7-6-5-4-3-2-10123χ
5           1-1
3          2 2
1         41 -3
-1        82  6
-3       95   -4
-5      107    3
-7     99     0
-9    810      -2
-11   59       4
-13  38        -5
-15 15         4
-17 3          -3
-191           1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-3 i=-1
r=-8 {\mathbb Z}
r=-7 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-6 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-5 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-4 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=-3 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=-2 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=-1 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=0 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{8}
r=1 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=2 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=3 {\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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K11a311.gif

K11a311

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K11a313