K11a327

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K11a326.gif

K11a326

K11a328.gif

K11a328

Contents

K11a327.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a327 at Knotilus!



Knot presentations

Planar diagram presentation X6271 X12,4,13,3 X18,6,19,5 X16,7,17,8 X14,10,15,9 X4,12,5,11 X20,14,21,13 X22,15,1,16 X2,18,3,17 X10,20,11,19 X8,21,9,22
Gauss code 1, -9, 2, -6, 3, -1, 4, -11, 5, -10, 6, -2, 7, -5, 8, -4, 9, -3, 10, -7, 11, -8
Dowker-Thistlethwaite code 6 12 18 16 14 4 20 22 2 10 8
A Braid Representative
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A Morse Link Presentation K11a327 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number \{1,2\}
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a327/ThurstonBennequinNumber
Hyperbolic Volume 19.2356
A-Polynomial See Data:K11a327/A-polynomial

[edit Notes for K11a327's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 3
Rasmussen s-Invariant -2

[edit Notes for K11a327's four dimensional invariants]

Polynomial invariants

Alexander polynomial 3 t^3-17 t^2+44 t-59+44 t^{-1} -17 t^{-2} +3 t^{-3}
Conway polynomial 3 z^6+z^4+3 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 187, 2 }
Jones polynomial q^9-5 q^8+11 q^7-19 q^6+26 q^5-30 q^4+31 q^3-26 q^2+20 q-12+5 q^{-1} - q^{-2}
HOMFLY-PT polynomial (db, data sources) z^6 a^{-2} +2 z^6 a^{-4} +5 z^4 a^{-4} -3 z^4 a^{-6} -z^4-z^2 a^{-2} +7 z^2 a^{-4} -4 z^2 a^{-6} +z^2 a^{-8} +3 a^{-4} -2 a^{-6}
Kauffman polynomial (db, data sources) 4 z^{10} a^{-4} +4 z^{10} a^{-6} +13 z^9 a^{-3} +23 z^9 a^{-5} +10 z^9 a^{-7} +17 z^8 a^{-2} +26 z^8 a^{-4} +19 z^8 a^{-6} +10 z^8 a^{-8} +12 z^7 a^{-1} -8 z^7 a^{-3} -36 z^7 a^{-5} -11 z^7 a^{-7} +5 z^7 a^{-9} -26 z^6 a^{-2} -71 z^6 a^{-4} -61 z^6 a^{-6} -20 z^6 a^{-8} +z^6 a^{-10} +5 z^6+a z^5-14 z^5 a^{-1} -16 z^5 a^{-3} -3 z^5 a^{-5} -11 z^5 a^{-7} -9 z^5 a^{-9} +14 z^4 a^{-2} +54 z^4 a^{-4} +50 z^4 a^{-6} +12 z^4 a^{-8} -z^4 a^{-10} -3 z^4+4 z^3 a^{-1} +15 z^3 a^{-3} +23 z^3 a^{-5} +16 z^3 a^{-7} +4 z^3 a^{-9} -4 z^2 a^{-2} -16 z^2 a^{-4} -15 z^2 a^{-6} -3 z^2 a^{-8} -z a^{-1} -4 z a^{-3} -8 z a^{-5} -5 z a^{-7} +3 a^{-4} +2 a^{-6}
The A2 invariant Data:K11a327/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a327/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: (3, 5)
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
12 40 72 174 26 480 \frac{2608}{3} \frac{448}{3} 136 288 800 2088 312 \frac{42751}{10} -\frac{1106}{15} \frac{9714}{5} \frac{11}{2} \frac{2751}{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=2 is the signature of K11a327. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-3-2-1012345678χ
19           11
17          4 -4
15         71 6
13        124  -8
11       147   7
9      1612    -4
7     1514     1
5    1116      5
3   915       -6
1  412        8
-1 18         -7
-3 4          4
-51           -1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=1 i=3
r=-3 {\mathbb Z}
r=-2 {\mathbb Z}^{4}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-1 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=0 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{9}
r=1 {\mathbb Z}^{15}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=2 {\mathbb Z}^{16}\oplus{\mathbb Z}_2^{15} {\mathbb Z}^{15}
r=3 {\mathbb Z}^{14}\oplus{\mathbb Z}_2^{16} {\mathbb Z}^{16}
r=4 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{14} {\mathbb Z}^{14}
r=5 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{12}
r=6 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=7 {\mathbb Z}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
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).

See/edit the Hoste-Thistlethwaite_Splice_Base (expert).

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K11a326.gif

K11a326

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K11a328