K11n4

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K11n3.gif

K11n3

K11n5.gif

K11n5

Contents

K11n4.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11n4 at Knotilus!


Knot K11n4.
A graph, knot K11n4.
A part of a knot and a part of a graph.

Knot presentations

Planar diagram presentation X4251 X8394 X10,6,11,5 X14,8,15,7 X2,9,3,10 X18,12,19,11 X6,14,7,13 X15,20,16,21 X12,18,13,17 X19,22,20,1 X21,16,22,17
Gauss code 1, -5, 2, -1, 3, -7, 4, -2, 5, -3, 6, -9, 7, -4, -8, 11, 9, -6, -10, 8, -11, 10
Dowker-Thistlethwaite code 4 8 10 14 2 18 6 -20 12 -22 -16
A Braid Representative
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A Morse Link Presentation K11n4 ML.gif

Three dimensional invariants

Symmetry type Chiral
Unknotting number \{1,2\}
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11n4/ThurstonBennequinNumber
Hyperbolic Volume 12.5531
A-Polynomial See Data:K11n4/A-polynomial

[edit Notes for K11n4's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus [0,3]
Rasmussen s-Invariant 0

[edit Notes for K11n4's four dimensional invariants]

Polynomial invariants

Alexander polynomial -t^3+5 t^2-11 t+15-11 t^{-1} +5 t^{-2} - t^{-3}
Conway polynomial -z^6-z^4+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 49, 0 }
Jones polynomial q^6-3 q^5+5 q^4-7 q^3+8 q^2-8 q+8-5 q^{-1} +3 q^{-2} - q^{-3}
HOMFLY-PT polynomial (db, data sources) -z^6 a^{-2} -4 z^4 a^{-2} +z^4 a^{-4} +2 z^4-a^2 z^2-6 z^2 a^{-2} +2 z^2 a^{-4} +5 z^2-a^2-3 a^{-2} + a^{-4} +4
Kauffman polynomial (db, data sources) z^9 a^{-1} +z^9 a^{-3} +5 z^8 a^{-2} +3 z^8 a^{-4} +2 z^8+a z^7+z^7 a^{-1} +3 z^7 a^{-3} +3 z^7 a^{-5} -15 z^6 a^{-2} -8 z^6 a^{-4} +z^6 a^{-6} -6 z^6-a z^5-8 z^5 a^{-1} -17 z^5 a^{-3} -10 z^5 a^{-5} +3 a^2 z^4+16 z^4 a^{-2} +3 z^4 a^{-4} -3 z^4 a^{-6} +13 z^4+a^3 z^3+5 a z^3+12 z^3 a^{-1} +16 z^3 a^{-3} +8 z^3 a^{-5} -3 a^2 z^2-10 z^2 a^{-2} -z^2 a^{-4} +2 z^2 a^{-6} -10 z^2-a^3 z-3 a z-5 z a^{-1} -5 z a^{-3} -2 z a^{-5} +a^2+3 a^{-2} + a^{-4} +4
The A2 invariant Data:K11n4/QuantumInvariant/A2/1,0
The G2 invariant Data:K11n4/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {9_27, K11n21, K11n172,}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {9_41, K11n21,}

Vassiliev invariants

V2 and V3: (0, -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
0 -8 0 0 8 0 \frac{112}{3} \frac{64}{3} 24 0 32 0 0 112 \frac{200}{3} -\frac{8}{3} \frac{64}{3} -16

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 K11n4. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-3-2-10123456χ
13         11
11        2 -2
9       31 2
7      42  -2
5     43   1
3    44    0
1   44     0
-1  25      3
-3 13       -2
-5 2        2
-71         -1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-1 i=1
r=-3 {\mathbb Z}
r=-2 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-1 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=0 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{4}
r=1 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=2 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=4 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=5 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
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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K11n3.gif

K11n3

K11n5.gif

K11n5