K11a212

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K11a211.gif

K11a211

K11a213.gif

K11a213

Contents

K11a212.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a212 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X12,4,13,3 X16,5,17,6 X22,8,1,7 X18,10,19,9 X2,12,3,11 X8,14,9,13 X6,15,7,16 X20,18,21,17 X10,20,11,19 X14,22,15,21
Gauss code 1, -6, 2, -1, 3, -8, 4, -7, 5, -10, 6, -2, 7, -11, 8, -3, 9, -5, 10, -9, 11, -4
Dowker-Thistlethwaite code 4 12 16 22 18 2 8 6 20 10 14
A Braid Representative
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A Morse Link Presentation K11a212 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number 2
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a212/ThurstonBennequinNumber
Hyperbolic Volume 17.7092
A-Polynomial See Data:K11a212/A-polynomial

[edit Notes for K11a212's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11a212's four dimensional invariants]

Polynomial invariants

Alexander polynomial -3 t^3+16 t^2-35 t+45-35 t^{-1} +16 t^{-2} -3 t^{-3}
Conway polynomial -3 z^6-2 z^4+2 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 153, 4 }
Jones polynomial -q^{11}+5 q^{10}-11 q^9+17 q^8-23 q^7+25 q^6-24 q^5+21 q^4-14 q^3+8 q^2-3 q+1
HOMFLY-PT polynomial (db, data sources) -z^6 a^{-4} -2 z^6 a^{-6} +z^4 a^{-2} -z^4 a^{-4} -5 z^4 a^{-6} +3 z^4 a^{-8} +2 z^2 a^{-2} +z^2 a^{-4} -4 z^2 a^{-6} +4 z^2 a^{-8} -z^2 a^{-10} + a^{-2} + a^{-4} - a^{-6}
Kauffman polynomial (db, data sources) 2 z^{10} a^{-6} +2 z^{10} a^{-8} +5 z^9 a^{-5} +13 z^9 a^{-7} +8 z^9 a^{-9} +5 z^8 a^{-4} +13 z^8 a^{-6} +21 z^8 a^{-8} +13 z^8 a^{-10} +3 z^7 a^{-3} -3 z^7 a^{-5} -13 z^7 a^{-7} +4 z^7 a^{-9} +11 z^7 a^{-11} +z^6 a^{-2} -9 z^6 a^{-4} -36 z^6 a^{-6} -48 z^6 a^{-8} -17 z^6 a^{-10} +5 z^6 a^{-12} -7 z^5 a^{-3} -9 z^5 a^{-5} -14 z^5 a^{-7} -28 z^5 a^{-9} -15 z^5 a^{-11} +z^5 a^{-13} -3 z^4 a^{-2} +5 z^4 a^{-4} +31 z^4 a^{-6} +31 z^4 a^{-8} +4 z^4 a^{-10} -4 z^4 a^{-12} +5 z^3 a^{-3} +12 z^3 a^{-5} +19 z^3 a^{-7} +17 z^3 a^{-9} +5 z^3 a^{-11} +3 z^2 a^{-2} -2 z^2 a^{-4} -11 z^2 a^{-6} -7 z^2 a^{-8} -z^2 a^{-10} -z a^{-3} -5 z a^{-5} -5 z a^{-7} -z a^{-9} - a^{-2} + a^{-4} + a^{-6}
The A2 invariant Data:K11a212/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a212/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, 3)
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 24 32 \frac{316}{3} \frac{92}{3} 192 464 96 120 \frac{256}{3} 288 \frac{2528}{3} \frac{736}{3} \frac{29311}{15} -\frac{2084}{15} \frac{53284}{45} \frac{449}{9} \frac{2191}{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=4 is the signature of K11a212. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-2-10123456789χ
23           1-1
21          4 4
19         71 -6
17        104  6
15       137   -6
13      1210    2
11     1213     1
9    912      -3
7   512       7
5  39        -6
3 16         5
1 2          -2
-11           1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=3 i=5
r=-2 {\mathbb Z}
r=-1 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=0 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{3}
r=1 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=2 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=3 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{12}
r=4 {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{12}
r=5 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{13} {\mathbb Z}^{13}
r=6 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=7 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=8 {\mathbb Z}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=9 {\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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K11a211

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K11a213