K11n178

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K11n177.gif

K11n177

K11n179.gif

K11n179

Contents

K11n178.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11n178 at Knotilus!



Knot presentations

Planar diagram presentation X6271 X3,11,4,10 X16,6,17,5 X18,7,19,8 X20,10,21,9 X11,5,12,4 X8,14,9,13 X2,16,3,15 X22,17,1,18 X14,19,15,20 X12,22,13,21
Gauss code 1, -8, -2, 6, 3, -1, 4, -7, 5, 2, -6, -11, 7, -10, 8, -3, 9, -4, 10, -5, 11, -9
Dowker-Thistlethwaite code 6 -10 16 18 20 -4 8 2 22 14 12
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart1.gifBraidPart1.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gif
BraidPart0.gifBraidPart2.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gif
A Morse Link Presentation K11n178 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:K11n178/ThurstonBennequinNumber
Hyperbolic Volume 16.9983
A-Polynomial See Data:K11n178/A-polynomial

[edit Notes for K11n178'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 K11n178's four dimensional invariants]

Polynomial invariants

Alexander polynomial 2 t^3-9 t^2+22 t-29+22 t^{-1} -9 t^{-2} +2 t^{-3}
Conway polynomial 2 z^6+3 z^4+4 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 95, 2 }
Jones polynomial q^9-5 q^8+9 q^7-13 q^6+16 q^5-16 q^4+15 q^3-11 q^2+7 q-2
HOMFLY-PT polynomial (db, data sources) 2 z^6 a^{-4} -2 z^4 a^{-2} +8 z^4 a^{-4} -3 z^4 a^{-6} -3 z^2 a^{-2} +12 z^2 a^{-4} -6 z^2 a^{-6} +z^2 a^{-8} - a^{-2} +5 a^{-4} -3 a^{-6}
Kauffman polynomial (db, data sources) 4 z^9 a^{-5} +4 z^9 a^{-7} +9 z^8 a^{-4} +17 z^8 a^{-6} +8 z^8 a^{-8} +6 z^7 a^{-3} +3 z^7 a^{-5} +2 z^7 a^{-7} +5 z^7 a^{-9} +z^6 a^{-2} -22 z^6 a^{-4} -44 z^6 a^{-6} -20 z^6 a^{-8} +z^6 a^{-10} -7 z^5 a^{-3} -20 z^5 a^{-5} -24 z^5 a^{-7} -11 z^5 a^{-9} +8 z^4 a^{-2} +31 z^4 a^{-4} +35 z^4 a^{-6} +11 z^4 a^{-8} -z^4 a^{-10} +3 z^3 a^{-1} +9 z^3 a^{-3} +16 z^3 a^{-5} +14 z^3 a^{-7} +4 z^3 a^{-9} -6 z^2 a^{-2} -20 z^2 a^{-4} -15 z^2 a^{-6} -z^2 a^{-8} -z a^{-1} -3 z a^{-3} -5 z a^{-5} -z a^{-7} +2 z a^{-9} + a^{-2} +5 a^{-4} +3 a^{-6}
The A2 invariant -2+3 q^{-2} -2 q^{-4} + q^{-6} +4 q^{-8} -2 q^{-10} +4 q^{-12} -2 q^{-14} +2 q^{-16} + q^{-18} -3 q^{-20} +2 q^{-22} -3 q^{-24} - q^{-26} + q^{-28}
The G2 invariant Data:K11n178/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: (4, 7)
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 56 128 \frac{824}{3} \frac{112}{3} 896 \frac{4496}{3} \frac{800}{3} 184 \frac{2048}{3} 1568 \frac{13184}{3} \frac{1792}{3} \frac{124142}{15} \frac{4192}{15} \frac{138608}{45} \frac{610}{9} \frac{6062}{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 K11n178. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-1012345678χ
19         11
17        4 -4
15       51 4
13      84  -4
11     85   3
9    88    0
7   78     -1
5  48      4
3 37       -4
1 5        5
-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}^{5}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{3}
r=1 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=2 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=3 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=4 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=5 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=6 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
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

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K11n177.gif

K11n177

K11n179.gif

K11n179