K11n13

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K11n12.gif

K11n12

K11n14.gif

K11n14

Contents

K11n13.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11n13 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X8394 X10,6,11,5 X16,8,17,7 X2,9,3,10 X11,19,12,18 X13,21,14,20 X6,16,7,15 X17,1,18,22 X19,13,20,12 X21,15,22,14
Gauss code 1, -5, 2, -1, 3, -8, 4, -2, 5, -3, -6, 10, -7, 11, 8, -4, -9, 6, -10, 7, -11, 9
Dowker-Thistlethwaite code 4 8 10 16 2 -18 -20 6 -22 -12 -14
A Braid Representative
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A Morse Link Presentation K11n13 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number 3
3-genus 4
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11n13/ThurstonBennequinNumber
Hyperbolic Volume 6.56631
A-Polynomial See Data:K11n13/A-polynomial

[edit Notes for K11n13's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 4
Rasmussen s-Invariant -6

[edit Notes for K11n13's four dimensional invariants]

Polynomial invariants

Alexander polynomial -t^4+3 t^3-2 t^2+t-1+ t^{-1} -2 t^{-2} +3 t^{-3} - t^{-4}
Conway polynomial -z^8-5 z^6-4 z^4+4 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 15, 6 }
Jones polynomial -q^{10}+q^9-q^8+2 q^7-2 q^6+2 q^5-2 q^4+2 q^3-q^2+q
HOMFLY-PT polynomial (db, data sources) -z^8 a^{-6} +z^6 a^{-4} -7 z^6 a^{-6} +z^6 a^{-8} +6 z^4 a^{-4} -16 z^4 a^{-6} +6 z^4 a^{-8} +10 z^2 a^{-4} -15 z^2 a^{-6} +10 z^2 a^{-8} -z^2 a^{-10} +4 a^{-4} -6 a^{-6} +5 a^{-8} -2 a^{-10}
Kauffman polynomial (db, data sources) z^9 a^{-5} +z^9 a^{-7} +z^8 a^{-4} +3 z^8 a^{-6} +2 z^8 a^{-8} -6 z^7 a^{-5} -5 z^7 a^{-7} +z^7 a^{-9} -7 z^6 a^{-4} -19 z^6 a^{-6} -12 z^6 a^{-8} +10 z^5 a^{-5} +6 z^5 a^{-7} -4 z^5 a^{-9} +16 z^4 a^{-4} +37 z^4 a^{-6} +23 z^4 a^{-8} +2 z^4 a^{-10} -4 z^3 a^{-5} -2 z^3 a^{-7} +3 z^3 a^{-9} +z^3 a^{-11} -14 z^2 a^{-4} -26 z^2 a^{-6} -19 z^2 a^{-8} -6 z^2 a^{-10} +z^2 a^{-12} +z a^{-7} -z a^{-9} -z a^{-11} +z a^{-13} +4 a^{-4} +6 a^{-6} +5 a^{-8} +2 a^{-10}
The A2 invariant  q^{-4} + q^{-6} + q^{-8} + q^{-10} - q^{-16} - q^{-20} + q^{-22} + q^{-24} + q^{-26} - q^{-30} - q^{-34}
The G2 invariant Data:K11n13/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {9_2,}

Vassiliev invariants

V2 and V3: (4, 10)
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
16 80 128 \frac{1448}{3} \frac{280}{3} 1280 \frac{8960}{3} \frac{1760}{3} 464 \frac{2048}{3} 3200 \frac{23168}{3} \frac{4480}{3} \frac{279542}{15} \frac{2392}{15} \frac{371768}{45} \frac{1306}{9} \frac{16742}{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=6 is the signature of K11n13. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-2-101234567χ
21         1-1
19          0
17       11 0
15      1   1
13     11   0
11    11    0
9   11     0
7  11      0
5 12       1
3          0
11         1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=5 i=7
r=-2 {\mathbb Z}
r=-1 {\mathbb Z}_2 {\mathbb Z}
r=0 {\mathbb Z}^{2} {\mathbb Z}
r=1 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=2 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=3 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=4 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=5 {\mathbb Z}_2 {\mathbb Z}
r=6 {\mathbb Z}
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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K11n12

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K11n14