K11a179

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K11a178.gif

K11a178

K11a180.gif

K11a180

Contents

K11a179.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a179 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X12,3,13,4 X14,6,15,5 X16,8,17,7 X18,10,19,9 X20,11,21,12 X2,13,3,14 X6,16,7,15 X8,18,9,17 X22,20,1,19 X10,21,11,22
Gauss code 1, -7, 2, -1, 3, -8, 4, -9, 5, -11, 6, -2, 7, -3, 8, -4, 9, -5, 10, -6, 11, -10
Dowker-Thistlethwaite code 4 12 14 16 18 20 2 6 8 22 10
A Braid Representative
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A Morse Link Presentation K11a179 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number 2
3-genus 4
Bridge index 2
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a179/ThurstonBennequinNumber
Hyperbolic Volume 10.0674
A-Polynomial See Data:K11a179/A-polynomial

[edit Notes for K11a179's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11a179's four dimensional invariants]

Polynomial invariants

Alexander polynomial t^4-5 t^3+9 t^2-9 t+9-9 t^{-1} +9 t^{-2} -5 t^{-3} + t^{-4}
Conway polynomial z^8+3 z^6-z^4-2 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 57, 4 }
Jones polynomial -q^9+3 q^8-4 q^7+6 q^6-8 q^5+8 q^4-8 q^3+7 q^2-5 q+4-2 q^{-1} + q^{-2}
HOMFLY-PT polynomial (db, data sources) z^8 a^{-4} -2 z^6 a^{-2} +6 z^6 a^{-4} -z^6 a^{-6} -10 z^4 a^{-2} +12 z^4 a^{-4} -4 z^4 a^{-6} +z^4-13 z^2 a^{-2} +10 z^2 a^{-4} -3 z^2 a^{-6} +4 z^2-4 a^{-2} +2 a^{-4} +3
Kauffman polynomial (db, data sources) z^{10} a^{-2} +z^{10} a^{-4} +2 z^9 a^{-1} +5 z^9 a^{-3} +3 z^9 a^{-5} -z^8 a^{-2} +2 z^8 a^{-4} +4 z^8 a^{-6} +z^8-11 z^7 a^{-1} -23 z^7 a^{-3} -8 z^7 a^{-5} +4 z^7 a^{-7} -15 z^6 a^{-2} -22 z^6 a^{-4} -9 z^6 a^{-6} +4 z^6 a^{-8} -6 z^6+18 z^5 a^{-1} +28 z^5 a^{-3} +2 z^5 a^{-5} -4 z^5 a^{-7} +4 z^5 a^{-9} +35 z^4 a^{-2} +31 z^4 a^{-4} +3 z^4 a^{-6} -2 z^4 a^{-8} +3 z^4 a^{-10} +12 z^4-9 z^3 a^{-1} -7 z^3 a^{-3} +z^3 a^{-5} -5 z^3 a^{-7} -3 z^3 a^{-9} +z^3 a^{-11} -23 z^2 a^{-2} -13 z^2 a^{-4} -z^2 a^{-6} -3 z^2 a^{-8} -2 z^2 a^{-10} -10 z^2-z a^{-3} +z a^{-5} +3 z a^{-7} +z a^{-9} +4 a^{-2} +2 a^{-4} +3
The A2 invariant q^6+q^4+1+ q^{-4} - q^{-8} -3 q^{-12} + q^{-14} + q^{-18} + q^{-20} + q^{-24} - q^{-26}
The G2 invariant q^{26}-q^{24}+4 q^{22}-5 q^{20}+6 q^{18}-4 q^{16}+11 q^{12}-18 q^{10}+24 q^8-21 q^6+10 q^4+8 q^2-25+39 q^{-2} -35 q^{-4} +23 q^{-6} -2 q^{-8} -17 q^{-10} +27 q^{-12} -29 q^{-14} +18 q^{-16} -4 q^{-18} -9 q^{-20} +15 q^{-22} -10 q^{-24} +12 q^{-28} -18 q^{-30} +17 q^{-32} -11 q^{-34} -6 q^{-36} +18 q^{-38} -32 q^{-40} +34 q^{-42} -25 q^{-44} +8 q^{-46} +9 q^{-48} -26 q^{-50} +31 q^{-52} -29 q^{-54} +15 q^{-56} -2 q^{-58} -10 q^{-60} +14 q^{-62} -10 q^{-64} +5 q^{-66} +3 q^{-68} -2 q^{-70} +2 q^{-72} +4 q^{-74} -6 q^{-76} +10 q^{-78} -6 q^{-80} +3 q^{-82} +3 q^{-84} -3 q^{-86} +6 q^{-88} -7 q^{-90} +7 q^{-92} -7 q^{-94} +9 q^{-96} -8 q^{-98} -3 q^{-100} +6 q^{-102} -11 q^{-104} +16 q^{-106} -16 q^{-108} +11 q^{-110} -2 q^{-112} -7 q^{-114} +13 q^{-116} -18 q^{-118} +14 q^{-120} -10 q^{-122} +5 q^{-124} +2 q^{-126} -8 q^{-128} +12 q^{-130} -9 q^{-132} +8 q^{-134} -3 q^{-136} + q^{-140} -4 q^{-142} +3 q^{-144} -2 q^{-146} + q^{-148}

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

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

Vassiliev invariants

V2 and V3: (-2, -2)
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 -16 32 \frac{116}{3} \frac{76}{3} 128 \frac{512}{3} \frac{32}{3} 48 -\frac{256}{3} 128 -\frac{928}{3} -\frac{608}{3} -\frac{2551}{15} -\frac{572}{5} -\frac{8764}{45} \frac{391}{9} -\frac{391}{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=4 is the signature of K11a179. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-4-3-2-101234567χ
19           1-1
17          2 2
15         21 -1
13        42  2
11       42   -2
9      44    0
7     44     0
5    34      -1
3   35       2
1  12        -1
-1 13         2
-3 1          -1
-51           1
Integral Khovanov Homology

(db, data source)

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

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K11a178

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K11a180