K11a348

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K11a347.gif

K11a347

K11a349.gif

K11a349

Contents

K11a348.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a348 at Knotilus!



Knot presentations

Planar diagram presentation X6271 X18,4,19,3 X16,5,17,6 X12,8,13,7 X4,10,5,9 X2,11,3,12 X20,14,21,13 X22,16,1,15 X10,18,11,17 X8,19,9,20 X14,22,15,21
Gauss code 1, -6, 2, -5, 3, -1, 4, -10, 5, -9, 6, -4, 7, -11, 8, -3, 9, -2, 10, -7, 11, -8
Dowker-Thistlethwaite code 6 18 16 12 4 2 20 22 10 8 14
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart1.gifBraidPart1.gif
BraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart2.gifBraidPart2.gif
A Morse Link Presentation K11a348 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number 2
3-genus 4
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a348/ThurstonBennequinNumber
Hyperbolic Volume 17.56
A-Polynomial See Data:K11a348/A-polynomial

[edit Notes for K11a348's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11a348's four dimensional invariants]

Polynomial invariants

Alexander polynomial t^4-7 t^3+19 t^2-29 t+33-29 t^{-1} +19 t^{-2} -7 t^{-3} + t^{-4}
Conway polynomial z^8+z^6-3 z^4+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 145, 4 }
Jones polynomial q^{10}-4 q^9+9 q^8-15 q^7+20 q^6-23 q^5+23 q^4-20 q^3+15 q^2-9 q+5- q^{-1}
HOMFLY-PT polynomial (db, data sources) z^8 a^{-4} -z^6 a^{-2} +4 z^6 a^{-4} -2 z^6 a^{-6} -2 z^4 a^{-2} +4 z^4 a^{-4} -6 z^4 a^{-6} +z^4 a^{-8} +2 z^2 a^{-2} -z^2 a^{-4} -3 z^2 a^{-6} +2 z^2 a^{-8} +3 a^{-2} -3 a^{-4} + a^{-6}
Kauffman polynomial (db, data sources) 4 z^{10} a^{-4} +4 z^{10} a^{-6} +8 z^9 a^{-3} +19 z^9 a^{-5} +11 z^9 a^{-7} +5 z^8 a^{-2} +z^8 a^{-4} +11 z^8 a^{-6} +15 z^8 a^{-8} +z^7 a^{-1} -27 z^7 a^{-3} -56 z^7 a^{-5} -14 z^7 a^{-7} +14 z^7 a^{-9} -16 z^6 a^{-2} -33 z^6 a^{-4} -50 z^6 a^{-6} -24 z^6 a^{-8} +9 z^6 a^{-10} -2 z^5 a^{-1} +26 z^5 a^{-3} +46 z^5 a^{-5} -5 z^5 a^{-7} -19 z^5 a^{-9} +4 z^5 a^{-11} +12 z^4 a^{-2} +34 z^4 a^{-4} +41 z^4 a^{-6} +11 z^4 a^{-8} -7 z^4 a^{-10} +z^4 a^{-12} -7 z^3 a^{-3} -12 z^3 a^{-5} +5 z^3 a^{-7} +9 z^3 a^{-9} -z^3 a^{-11} +2 z^2 a^{-2} -2 z^2 a^{-4} -9 z^2 a^{-6} -3 z^2 a^{-8} +2 z^2 a^{-10} +z a^{-3} +z a^{-5} -z a^{-7} -z a^{-9} -3 a^{-2} -3 a^{-4} - a^{-6}
The A2 invariant -q^2+3- q^{-2} +3 q^{-4} +2 q^{-6} -3 q^{-8} +4 q^{-10} -6 q^{-12} +2 q^{-14} - q^{-16} - q^{-18} +4 q^{-20} -3 q^{-22} +2 q^{-24} - q^{-26} - q^{-28} + q^{-30}
The G2 invariant Data:K11a348/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: (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 24 0 \frac{304}{3} \frac{352}{3} 24 0 32 0 0 432 312 152 -16 0

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 K11a348. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-3-2-1012345678χ
21           11
19          3 -3
17         61 5
15        93  -6
13       116   5
11      129    -3
9     1111     0
7    912      3
5   611       -5
3  410        6
1 15         -4
-1 4          4
-31           -1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=3 i=5
r=-3 {\mathbb Z}
r=-2 {\mathbb Z}^{4}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-1 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=0 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{6}
r=1 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=2 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=3 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{12}
r=4 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=5 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=6 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=7 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=8 {\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).

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K11a347.gif

K11a347

K11a349.gif

K11a349