K11a342

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K11a341.gif

K11a341

K11a343.gif

K11a343

Contents

K11a342.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a342 at Knotilus!



Knot presentations

Planar diagram presentation X6271 X14,4,15,3 X16,6,17,5 X22,8,1,7 X20,10,21,9 X18,12,19,11 X2,14,3,13 X4,16,5,15 X12,18,13,17 X10,20,11,19 X8,22,9,21
Gauss code 1, -7, 2, -8, 3, -1, 4, -11, 5, -10, 6, -9, 7, -2, 8, -3, 9, -6, 10, -5, 11, -4
Dowker-Thistlethwaite code 6 14 16 22 20 18 2 4 12 10 8
A Braid Representative
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A Morse Link Presentation K11a342 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number 2
3-genus 2
Bridge index 2
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a342/ThurstonBennequinNumber
Hyperbolic Volume 5.83875
A-Polynomial See Data:K11a342/A-polynomial

[edit Notes for K11a342's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11a342's four dimensional invariants]

Polynomial invariants

Alexander polynomial 4 t^2-7 t+7-7 t^{-1} +4 t^{-2}
Conway polynomial 4 z^4+9 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 29, 4 }
Jones polynomial -q^{13}+q^{12}-2 q^{11}+3 q^{10}-3 q^9+4 q^8-4 q^7+4 q^6-3 q^5+2 q^4-q^3+q^2
HOMFLY-PT polynomial (db, data sources) z^4 a^{-4} +z^4 a^{-6} +z^4 a^{-8} +z^4 a^{-10} +3 z^2 a^{-4} +2 z^2 a^{-6} +2 z^2 a^{-8} +3 z^2 a^{-10} -z^2 a^{-12} + a^{-4} +2 a^{-10} -2 a^{-12}
Kauffman polynomial (db, data sources) z^{10} a^{-10} +z^{10} a^{-12} +z^9 a^{-9} +2 z^9 a^{-11} +z^9 a^{-13} +z^8 a^{-8} -7 z^8 a^{-10} -7 z^8 a^{-12} +z^8 a^{-14} +z^7 a^{-7} -5 z^7 a^{-9} -12 z^7 a^{-11} -5 z^7 a^{-13} +z^7 a^{-15} +z^6 a^{-6} -4 z^6 a^{-8} +20 z^6 a^{-10} +20 z^6 a^{-12} -5 z^6 a^{-14} +z^5 a^{-5} -3 z^5 a^{-7} +9 z^5 a^{-9} +26 z^5 a^{-11} +7 z^5 a^{-13} -6 z^5 a^{-15} +z^4 a^{-4} -2 z^4 a^{-6} +6 z^4 a^{-8} -26 z^4 a^{-10} -29 z^4 a^{-12} +6 z^4 a^{-14} -2 z^3 a^{-5} +4 z^3 a^{-7} -4 z^3 a^{-9} -24 z^3 a^{-11} -4 z^3 a^{-13} +10 z^3 a^{-15} -3 z^2 a^{-4} +z^2 a^{-6} +13 z^2 a^{-10} +16 z^2 a^{-12} -z^2 a^{-14} +5 z a^{-11} +z a^{-13} -4 z a^{-15} + a^{-4} -2 a^{-10} -2 a^{-12}
The A2 invariant Data:K11a342/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a342/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: (9, 29)
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
36 232 648 1818 286 8352 \frac{47728}{3} \frac{8224}{3} 2344 7776 26912 65448 10296 \frac{1407693}{10} \frac{5642}{15} \frac{886786}{15} \frac{8851}{6} \frac{79053}{10}

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 K11a342. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
01234567891011χ
27           1-1
25            0
23         21 -1
21        1   1
19       22   0
17      21    1
15     22     0
13    22      0
11   12       1
9  12        -1
7  1         1
511          0
31           1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=3 i=5
r=0 {\mathbb Z} {\mathbb Z}
r=1 {\mathbb Z}
r=2 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=3 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=4 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=5 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=6 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=7 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=8 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=9 {\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=10 {\mathbb Z}
r=11 {\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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K11a341.gif

K11a341

K11a343.gif

K11a343