K11a80

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K11a79.gif

K11a79

K11a81.gif

K11a81

Contents

K11a80.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a80 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X10,3,11,4 X12,6,13,5 X14,7,15,8 X22,10,1,9 X2,11,3,12 X18,14,19,13 X20,15,21,16 X6,17,7,18 X8,20,9,19 X16,21,17,22
Gauss code 1, -6, 2, -1, 3, -9, 4, -10, 5, -2, 6, -3, 7, -4, 8, -11, 9, -7, 10, -8, 11, -5
Dowker-Thistlethwaite code 4 10 12 14 22 2 18 20 6 8 16
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
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BraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gif
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A Morse Link Presentation K11a80 ML.gif

Three dimensional invariants

Symmetry type Chiral
Unknotting number 1
3-genus 4
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a80/ThurstonBennequinNumber
Hyperbolic Volume 16.5161
A-Polynomial See Data:K11a80/A-polynomial

[edit Notes for K11a80's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 4
Rasmussen s-Invariant 0

[edit Notes for K11a80's four dimensional invariants]

Polynomial invariants

Alexander polynomial t^4-6 t^3+16 t^2-28 t+35-28 t^{-1} +16 t^{-2} -6 t^{-3} + t^{-4}
Conway polynomial z^8+2 z^6-2 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 137, 0 }
Jones polynomial -q^5+4 q^4-8 q^3+14 q^2-19 q+22-22 q^{-1} +19 q^{-2} -14 q^{-3} +9 q^{-4} -4 q^{-5} + q^{-6}
HOMFLY-PT polynomial (db, data sources) z^8-2 a^2 z^6-z^6 a^{-2} +5 z^6+a^4 z^4-7 a^2 z^4-3 z^4 a^{-2} +9 z^4+2 a^4 z^2-7 a^2 z^2-2 z^2 a^{-2} +5 z^2+a^4-a^2+ a^{-2}
Kauffman polynomial (db, data sources) 2 a^2 z^{10}+2 z^{10}+6 a^3 z^9+12 a z^9+6 z^9 a^{-1} +7 a^4 z^8+12 a^2 z^8+8 z^8 a^{-2} +13 z^8+4 a^5 z^7-7 a^3 z^7-19 a z^7-z^7 a^{-1} +7 z^7 a^{-3} +a^6 z^6-16 a^4 z^6-38 a^2 z^6-9 z^6 a^{-2} +4 z^6 a^{-4} -34 z^6-9 a^5 z^5-7 a^3 z^5+a z^5-12 z^5 a^{-1} -10 z^5 a^{-3} +z^5 a^{-5} -2 a^6 z^4+10 a^4 z^4+33 a^2 z^4-6 z^4 a^{-4} +27 z^4+5 a^5 z^3+8 a^3 z^3+11 a z^3+13 z^3 a^{-1} +4 z^3 a^{-3} -z^3 a^{-5} +a^6 z^2-4 a^4 z^2-11 a^2 z^2+3 z^2 a^{-2} +2 z^2 a^{-4} -5 z^2-a^5 z-2 a^3 z-4 a z-4 z a^{-1} -z a^{-3} +a^4+a^2- a^{-2}
The A2 invariant q^{18}-q^{16}+2 q^{12}-3 q^{10}+4 q^8-q^6-q^4+2 q^2-5+4 q^{-2} -3 q^{-4} +2 q^{-6} +3 q^{-8} -2 q^{-10} +2 q^{-12} - q^{-14}
The G2 invariant q^{94}-3 q^{92}+8 q^{90}-16 q^{88}+22 q^{86}-24 q^{84}+12 q^{82}+20 q^{80}-66 q^{78}+122 q^{76}-163 q^{74}+153 q^{72}-74 q^{70}-81 q^{68}+278 q^{66}-435 q^{64}+489 q^{62}-371 q^{60}+80 q^{58}+297 q^{56}-636 q^{54}+791 q^{52}-674 q^{50}+309 q^{48}+174 q^{46}-588 q^{44}+763 q^{42}-624 q^{40}+243 q^{38}+218 q^{36}-548 q^{34}+593 q^{32}-336 q^{30}-117 q^{28}+568 q^{26}-804 q^{24}+717 q^{22}-313 q^{20}-267 q^{18}+803 q^{16}-1097 q^{14}+1025 q^{12}-605 q^{10}-21 q^8+624 q^6-994 q^4+998 q^2-656+120 q^{-2} +386 q^{-4} -666 q^{-6} +614 q^{-8} -284 q^{-10} -153 q^{-12} +499 q^{-14} -586 q^{-16} +384 q^{-18} +11 q^{-20} -425 q^{-22} +690 q^{-24} -694 q^{-26} +460 q^{-28} -82 q^{-30} -296 q^{-32} +542 q^{-34} -600 q^{-36} +486 q^{-38} -256 q^{-40} +10 q^{-42} +184 q^{-44} -289 q^{-46} +294 q^{-48} -230 q^{-50} +132 q^{-52} -31 q^{-54} -46 q^{-56} +86 q^{-58} -95 q^{-60} +77 q^{-62} -46 q^{-64} +20 q^{-66} +3 q^{-68} -14 q^{-70} +15 q^{-72} -13 q^{-74} +7 q^{-76} -3 q^{-78} + q^{-80}

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {K11a270,}

Vassiliev invariants

V2 and V3: (-2, 1)
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 8 32 \frac{116}{3} \frac{52}{3} -64 -\frac{400}{3} -\frac{160}{3} -24 -\frac{256}{3} 32 -\frac{928}{3} -\frac{416}{3} -\frac{2311}{15} -\frac{796}{15} -\frac{2884}{45} \frac{199}{9} -\frac{151}{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=0 is the signature of K11a80. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-6-5-4-3-2-1012345χ
11           1-1
9          3 3
7         51 -4
5        93  6
3       105   -5
1      129    3
-1     1111     0
-3    811      -3
-5   611       5
-7  38        -5
-9 16         5
-11 3          -3
-131           1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-1 i=1
r=-6 {\mathbb Z}
r=-5 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-4 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-3 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=-2 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=-1 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=0 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{12}
r=1 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=2 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=4 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=5 {\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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K11a79.gif

K11a79

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K11a81