K11a213

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K11a212.gif

K11a212

K11a214.gif

K11a214

Contents

K11a213.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a213 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X12,4,13,3 X16,6,17,5 X22,8,1,7 X18,10,19,9 X2,12,3,11 X20,13,21,14 X10,16,11,15 X6,18,7,17 X14,19,15,20 X8,22,9,21
Gauss code 1, -6, 2, -1, 3, -9, 4, -11, 5, -8, 6, -2, 7, -10, 8, -3, 9, -5, 10, -7, 11, -4
Dowker-Thistlethwaite code 4 12 16 22 18 2 20 10 6 14 8
A Braid Representative
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A Morse Link Presentation K11a213 ML.gif

Three dimensional invariants

Symmetry type Chiral
Unknotting number 2
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a213/ThurstonBennequinNumber
Hyperbolic Volume 17.5099
A-Polynomial See Data:K11a213/A-polynomial

[edit Notes for K11a213'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 K11a213's four dimensional invariants]

Polynomial invariants

Alexander polynomial -3 t^3+16 t^2-33 t+41-33 t^{-1} +16 t^{-2} -3 t^{-3}
Conway polynomial -3 z^6-2 z^4+4 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 145, 4 }
Jones polynomial -q^{11}+4 q^{10}-10 q^9+16 q^8-21 q^7+24 q^6-23 q^5+20 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} -3 z^2 a^{-6} +5 z^2 a^{-8} -z^2 a^{-10} + a^{-2} + a^{-8} - a^{-10}
Kauffman polynomial (db, data sources) 2 z^{10} a^{-6} +2 z^{10} a^{-8} +5 z^9 a^{-5} +12 z^9 a^{-7} +7 z^9 a^{-9} +5 z^8 a^{-4} +11 z^8 a^{-6} +17 z^8 a^{-8} +11 z^8 a^{-10} +3 z^7 a^{-3} -4 z^7 a^{-5} -13 z^7 a^{-7} +3 z^7 a^{-9} +9 z^7 a^{-11} +z^6 a^{-2} -9 z^6 a^{-4} -29 z^6 a^{-6} -39 z^6 a^{-8} -16 z^6 a^{-10} +4 z^6 a^{-12} -7 z^5 a^{-3} -6 z^5 a^{-5} -9 z^5 a^{-7} -25 z^5 a^{-9} -14 z^5 a^{-11} +z^5 a^{-13} -3 z^4 a^{-2} +4 z^4 a^{-4} +20 z^4 a^{-6} +26 z^4 a^{-8} +9 z^4 a^{-10} -4 z^4 a^{-12} +5 z^3 a^{-3} +7 z^3 a^{-5} +12 z^3 a^{-7} +21 z^3 a^{-9} +10 z^3 a^{-11} -z^3 a^{-13} +3 z^2 a^{-2} -5 z^2 a^{-6} -7 z^2 a^{-8} -4 z^2 a^{-10} +z^2 a^{-12} -z a^{-3} -3 z a^{-5} -3 z a^{-7} -5 z a^{-9} -4 z a^{-11} - a^{-2} + a^{-8} + a^{-10}
The A2 invariant Data:K11a213/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a213/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: (4, 10)
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
16 80 128 \frac{1352}{3} \frac{232}{3} 1280 \frac{8288}{3} \frac{1376}{3} 464 \frac{2048}{3} 3200 \frac{21632}{3} \frac{3712}{3} \frac{257702}{15} -\frac{6968}{15} \frac{353288}{45} \frac{2506}{9} \frac{16022}{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 K11a213. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-2-10123456789χ
23           1-1
21          3 3
19         71 -6
17        93  6
15       127   -5
13      129    3
11     1112     1
9    912      -3
7   511       6
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}^{11}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=3 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=4 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{12}
r=5 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{12}
r=6 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=7 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=8 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
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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K11a212.gif

K11a212

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K11a214