K11a12

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K11a11.gif

K11a11

K11a13.gif

K11a13

Contents

K11a12.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a12 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X8394 X10,6,11,5 X16,7,17,8 X2,9,3,10 X20,12,21,11 X22,14,1,13 X18,16,19,15 X6,17,7,18 X14,20,15,19 X12,22,13,21
Gauss code 1, -5, 2, -1, 3, -9, 4, -2, 5, -3, 6, -11, 7, -10, 8, -4, 9, -8, 10, -6, 11, -7
Dowker-Thistlethwaite code 4 8 10 16 2 20 22 18 6 14 12
A Braid Representative
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A Morse Link Presentation K11a12 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number 1
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a12/ThurstonBennequinNumber
Hyperbolic Volume 14.2618
A-Polynomial See Data:K11a12/A-polynomial

[edit Notes for K11a12'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 K11a12's four dimensional invariants]

Polynomial invariants

Alexander polynomial 2 t^3-11 t^2+24 t-29+24 t^{-1} -11 t^{-2} +2 t^{-3}
Conway polynomial 2 z^6+z^4-2 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 103, 2 }
Jones polynomial -q^8+4 q^7-8 q^6+12 q^5-15 q^4+17 q^3-16 q^2+13 q-9+5 q^{-1} -2 q^{-2} + q^{-3}
HOMFLY-PT polynomial (db, data sources) z^6 a^{-2} +z^6 a^{-4} +2 z^4 a^{-2} +2 z^4 a^{-4} -z^4 a^{-6} -2 z^4+a^2 z^2+2 z^2 a^{-2} +z^2 a^{-4} -z^2 a^{-6} -5 z^2+2 a^2+2 a^{-2} -3
Kauffman polynomial (db, data sources) z^{10} a^{-2} +z^{10} a^{-4} +2 z^9 a^{-1} +6 z^9 a^{-3} +4 z^9 a^{-5} +3 z^8 a^{-2} +8 z^8 a^{-4} +7 z^8 a^{-6} +2 z^8+2 a z^7-z^7 a^{-1} -11 z^7 a^{-3} -z^7 a^{-5} +7 z^7 a^{-7} +a^2 z^6-5 z^6 a^{-2} -18 z^6 a^{-4} -11 z^6 a^{-6} +4 z^6 a^{-8} -z^6-6 a z^5-2 z^5 a^{-1} +13 z^5 a^{-3} -4 z^5 a^{-5} -12 z^5 a^{-7} +z^5 a^{-9} -4 a^2 z^4-z^4 a^{-2} +15 z^4 a^{-4} +5 z^4 a^{-6} -6 z^4 a^{-8} -9 z^4+5 a z^3-z^3 a^{-1} -10 z^3 a^{-3} +z^3 a^{-5} +4 z^3 a^{-7} -z^3 a^{-9} +5 a^2 z^2+4 z^2 a^{-2} -4 z^2 a^{-4} -z^2 a^{-6} +z^2 a^{-8} +11 z^2-a z+z a^{-1} +3 z a^{-3} +z a^{-5} -2 a^2-2 a^{-2} -3
The A2 invariant Data:K11a12/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a12/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a141,}

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

Vassiliev invariants

V2 and V3: (-2, 0)
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
-8 0 32 \frac{116}{3} \frac{28}{3} 0 -32 -64 32 -\frac{256}{3} 0 -\frac{928}{3} -\frac{224}{3} -\frac{4951}{15} -\frac{4676}{15} \frac{5876}{45} \frac{7}{9} \frac{569}{15}

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 K11a12. 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         51 -4
11        73  4
9       85   -3
7      97    2
5     78     1
3    69      -3
1   48       4
-1  15        -4
-3 14         3
-5 1          -1
-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}\oplus{\mathbb Z}_2 {\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}^{8}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{6}
r=1 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=2 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=3 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=4 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=5 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
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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K11a11

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K11a13