K11a314

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K11a313.gif

K11a313

K11a315.gif

K11a315

Contents

K11a314.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a314 at Knotilus!



Knot presentations

Planar diagram presentation X6271 X12,4,13,3 X16,5,17,6 X18,8,19,7 X14,10,15,9 X4,12,5,11 X20,13,21,14 X22,16,1,15 X2,17,3,18 X8,20,9,19 X10,21,11,22
Gauss code 1, -9, 2, -6, 3, -1, 4, -10, 5, -11, 6, -2, 7, -5, 8, -3, 9, -4, 10, -7, 11, -8
Dowker-Thistlethwaite code 6 12 16 18 14 4 20 22 2 8 10
A Braid Representative
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A Morse Link Presentation K11a314 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number \{1,2\}
3-genus 4
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a314/ThurstonBennequinNumber
Hyperbolic Volume 18.9578
A-Polynomial See Data:K11a314/A-polynomial

[edit Notes for K11a314's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11a314's four dimensional invariants]

Polynomial invariants

Alexander polynomial -t^4+7 t^3-21 t^2+36 t-41+36 t^{-1} -21 t^{-2} +7 t^{-3} - t^{-4}
Conway polynomial -z^8-z^6+z^4-z^2+1
2nd Alexander ideal (db, data sources) \{3,t+1\}
Determinant and Signature { 171, 2 }
Jones polynomial -q^8+4 q^7-9 q^6+17 q^5-24 q^4+27 q^3-28 q^2+25 q-18+12 q^{-1} -5 q^{-2} + q^{-3}
HOMFLY-PT polynomial (db, data sources) -z^8 a^{-2} -4 z^6 a^{-2} +2 z^6 a^{-4} +z^6-6 z^4 a^{-2} +6 z^4 a^{-4} -z^4 a^{-6} +2 z^4-5 z^2 a^{-2} +5 z^2 a^{-4} -2 z^2 a^{-6} +z^2-2 a^{-2} + a^{-4} +2
Kauffman polynomial (db, data sources) 4 z^{10} a^{-2} +4 z^{10} a^{-4} +11 z^9 a^{-1} +21 z^9 a^{-3} +10 z^9 a^{-5} +15 z^8 a^{-2} +15 z^8 a^{-4} +11 z^8 a^{-6} +11 z^8+5 a z^7-19 z^7 a^{-1} -41 z^7 a^{-3} -9 z^7 a^{-5} +8 z^7 a^{-7} +a^2 z^6-51 z^6 a^{-2} -44 z^6 a^{-4} -14 z^6 a^{-6} +4 z^6 a^{-8} -24 z^6-8 a z^5+5 z^5 a^{-1} +21 z^5 a^{-3} -3 z^5 a^{-5} -10 z^5 a^{-7} +z^5 a^{-9} -a^2 z^4+38 z^4 a^{-2} +34 z^4 a^{-4} +6 z^4 a^{-6} -5 z^4 a^{-8} +14 z^4+2 a z^3+8 z^3 a^{-5} +5 z^3 a^{-7} -z^3 a^{-9} -10 z^2 a^{-2} -8 z^2 a^{-4} +z^2 a^{-6} +2 z^2 a^{-8} -3 z^2-z a^{-1} -4 z a^{-3} -4 z a^{-5} -z a^{-7} +2 a^{-2} + a^{-4} +2
The A2 invariant q^8-3 q^6+4 q^4-q^2+1+5 q^{-2} -6 q^{-4} +5 q^{-6} -5 q^{-8} + q^{-10} + q^{-12} -4 q^{-14} +5 q^{-16} -2 q^{-18} + q^{-20} + q^{-22} - q^{-24}
The G2 invariant Data:K11a314/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: (-1, -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
-4 -8 8 -\frac{14}{3} -\frac{10}{3} 32 \frac{16}{3} -\frac{128}{3} 24 -\frac{32}{3} 32 \frac{56}{3} \frac{40}{3} -\frac{31}{30} -\frac{2858}{15} \frac{4858}{45} \frac{703}{18} \frac{929}{30}

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=2 is the signature of K11a314. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-4-3-2-101234567χ
17           1-1
15          3 3
13         61 -5
11        113  8
9       136   -7
7      1411    3
5     1413     -1
3    1114      -3
1   815       7
-1  410        -6
-3 18         7
-5 4          -4
-71           1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=1 i=3
r=-4 {\mathbb Z}
r=-3 {\mathbb Z}^{4}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-2 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-1 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=0 {\mathbb Z}^{15}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{11}
r=1 {\mathbb Z}^{14}\oplus{\mathbb Z}_2^{14} {\mathbb Z}^{14}
r=2 {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{14} {\mathbb Z}^{14}
r=3 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{13} {\mathbb Z}^{13}
r=4 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=5 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=6 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=7 {\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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K11a313.gif

K11a313

K11a315.gif

K11a315