K11n71

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K11n70.gif

K11n70

K11n72.gif

K11n72

Contents

K11n71.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11n71 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X8493 X14,6,15,5 X2837 X9,20,10,21 X16,12,17,11 X6,14,7,13 X18,16,19,15 X12,18,13,17 X19,22,20,1 X21,10,22,11
Gauss code 1, -4, 2, -1, 3, -7, 4, -2, -5, 11, 6, -9, 7, -3, 8, -6, 9, -8, -10, 5, -11, 10
Dowker-Thistlethwaite code 4 8 14 2 -20 16 6 18 12 -22 -10
A Braid Representative
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A Morse Link Presentation K11n71 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number \{1,2\}
3-genus 3
Bridge index 4
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11n71/ThurstonBennequinNumber
Hyperbolic Volume 13.5931
A-Polynomial See Data:K11n71/A-polynomial

[edit Notes for K11n71's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 1
Rasmussen s-Invariant -2

[edit Notes for K11n71's four dimensional invariants]

Polynomial invariants

Alexander polynomial 2 t^3-7 t^2+14 t-17+14 t^{-1} -7 t^{-2} +2 t^{-3}
Conway polynomial 2 z^6+5 z^4+4 z^2+1
2nd Alexander ideal (db, data sources) \left\{t^2-t+1\right\}
Determinant and Signature { 63, 2 }
Jones polynomial q^9-3 q^8+5 q^7-9 q^6+10 q^5-10 q^4+11 q^3-7 q^2+5 q-2
HOMFLY-PT polynomial (db, data sources) 2 z^6 a^{-4} -2 z^4 a^{-2} +10 z^4 a^{-4} -3 z^4 a^{-6} -5 z^2 a^{-2} +18 z^2 a^{-4} -10 z^2 a^{-6} +z^2 a^{-8} -3 a^{-2} +11 a^{-4} -9 a^{-6} +2 a^{-8}
Kauffman polynomial (db, data sources) z^9 a^{-5} +z^9 a^{-7} +4 z^8 a^{-4} +7 z^8 a^{-6} +3 z^8 a^{-8} +4 z^7 a^{-3} +8 z^7 a^{-5} +7 z^7 a^{-7} +3 z^7 a^{-9} +z^6 a^{-2} -10 z^6 a^{-4} -18 z^6 a^{-6} -6 z^6 a^{-8} +z^6 a^{-10} -10 z^5 a^{-3} -31 z^5 a^{-5} -31 z^5 a^{-7} -10 z^5 a^{-9} +4 z^4 a^{-2} +19 z^4 a^{-4} +17 z^4 a^{-6} -z^4 a^{-8} -3 z^4 a^{-10} +3 z^3 a^{-1} +19 z^3 a^{-3} +42 z^3 a^{-5} +36 z^3 a^{-7} +10 z^3 a^{-9} -6 z^2 a^{-2} -20 z^2 a^{-4} -15 z^2 a^{-6} +z^2 a^{-8} +2 z^2 a^{-10} -3 z a^{-1} -11 z a^{-3} -21 z a^{-5} -17 z a^{-7} -4 z a^{-9} +3 a^{-2} +11 a^{-4} +9 a^{-6} +2 a^{-8}
The A2 invariant Data:K11n71/QuantumInvariant/A2/1,0
The G2 invariant Data:K11n71/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {10_65, 10_77, K11n75,}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {K11n75,}

Vassiliev invariants

V2 and V3: (4, 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
16 40 128 \frac{536}{3} \frac{64}{3} 640 \frac{2416}{3} \frac{352}{3} 104 \frac{2048}{3} 800 \frac{8576}{3} \frac{1024}{3} \frac{58622}{15} \frac{3632}{15} \frac{53888}{45} \frac{370}{9} \frac{1982}{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 K11n71. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-1012345678χ
19         11
17        2 -2
15       31 2
13      62  -4
11     43   1
9    66    0
7   54     1
5  26      4
3 35       -2
1 3        3
-12         -2
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=1 i=3
r=-1 {\mathbb Z}^{2}
r=0 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{3}
r=1 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=2 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=3 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=4 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=5 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=6 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=7 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
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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K11n70.gif

K11n70

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K11n72