K11a28

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K11a27.gif

K11a27

K11a29.gif

K11a29

Contents

K11a28.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a28 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X8394 X12,6,13,5 X16,7,17,8 X2,9,3,10 X18,11,19,12 X20,14,21,13 X22,16,1,15 X10,17,11,18 X6,19,7,20 X14,22,15,21
Gauss code 1, -5, 2, -1, 3, -10, 4, -2, 5, -9, 6, -3, 7, -11, 8, -4, 9, -6, 10, -7, 11, -8
Dowker-Thistlethwaite code 4 8 12 16 2 18 20 22 10 6 14
A Braid Representative
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A Morse Link Presentation K11a28 ML.gif

Three dimensional invariants

Symmetry type Chiral
Unknotting number \{1,2\}
3-genus 4
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a28/ThurstonBennequinNumber
Hyperbolic Volume 15.8799
A-Polynomial See Data:K11a28/A-polynomial

[edit Notes for K11a28's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus [0,4]
Rasmussen s-Invariant 0

[edit Notes for K11a28's four dimensional invariants]

Polynomial invariants

Alexander polynomial t^4-6 t^3+15 t^2-24 t+29-24 t^{-1} +15 t^{-2} -6 t^{-3} + t^{-4}
Conway polynomial z^8+2 z^6-z^4-2 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 121, 0 }
Jones polynomial -q^5+4 q^4-8 q^3+13 q^2-17 q+20-19 q^{-1} +16 q^{-2} -12 q^{-3} +7 q^{-4} -3 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^4-8 a^2 z^4-3 z^4 a^{-2} +9 z^4+3 a^4 z^2-10 a^2 z^2-2 z^2 a^{-2} +7 z^2+2 a^4-4 a^2+3
Kauffman polynomial (db, data sources) 2 a^2 z^{10}+2 z^{10}+5 a^3 z^9+11 a z^9+6 z^9 a^{-1} +5 a^4 z^8+6 a^2 z^8+8 z^8 a^{-2} +9 z^8+3 a^5 z^7-11 a^3 z^7-28 a z^7-7 z^7 a^{-1} +7 z^7 a^{-3} +a^6 z^6-13 a^4 z^6-28 a^2 z^6-13 z^6 a^{-2} +4 z^6 a^{-4} -31 z^6-8 a^5 z^5+9 a^3 z^5+31 a z^5+2 z^5 a^{-1} -11 z^5 a^{-3} +z^5 a^{-5} -3 a^6 z^4+11 a^4 z^4+38 a^2 z^4+7 z^4 a^{-2} -6 z^4 a^{-4} +37 z^4+5 a^5 z^3-6 a^3 z^3-15 a z^3+3 z^3 a^{-3} -z^3 a^{-5} +2 a^6 z^2-7 a^4 z^2-22 a^2 z^2-2 z^2 a^{-2} +z^2 a^{-4} -16 z^2-a^5 z+a^3 z+3 a z+z a^{-1} +2 a^4+4 a^2+3
The A2 invariant q^{18}+2 q^{12}-3 q^{10}+2 q^8-2 q^6-2 q^4+3 q^2-3+5 q^{-2} -2 q^{-4} + q^{-6} +2 q^{-8} -2 q^{-10} +2 q^{-12} - q^{-14}
The G2 invariant q^{94}-2 q^{92}+5 q^{90}-9 q^{88}+11 q^{86}-12 q^{84}+5 q^{82}+11 q^{80}-33 q^{78}+61 q^{76}-81 q^{74}+79 q^{72}-47 q^{70}-21 q^{68}+120 q^{66}-215 q^{64}+274 q^{62}-251 q^{60}+121 q^{58}+88 q^{56}-314 q^{54}+476 q^{52}-483 q^{50}+327 q^{48}-38 q^{46}-283 q^{44}+499 q^{42}-523 q^{40}+337 q^{38}-25 q^{36}-273 q^{34}+422 q^{32}-359 q^{30}+115 q^{28}+202 q^{26}-448 q^{24}+502 q^{22}-341 q^{20}-3 q^{18}+378 q^{16}-653 q^{14}+716 q^{12}-528 q^{10}+163 q^8+264 q^6-606 q^4+735 q^2-612+288 q^{-2} +107 q^{-4} -413 q^{-6} +520 q^{-8} -381 q^{-10} +100 q^{-12} +206 q^{-14} -387 q^{-16} +362 q^{-18} -157 q^{-20} -144 q^{-22} +397 q^{-24} -489 q^{-26} +401 q^{-28} -161 q^{-30} -121 q^{-32} +341 q^{-34} -440 q^{-36} +397 q^{-38} -250 q^{-40} +59 q^{-42} +109 q^{-44} -215 q^{-46} +243 q^{-48} -203 q^{-50} +132 q^{-52} -44 q^{-54} -30 q^{-56} +72 q^{-58} -88 q^{-60} +73 q^{-62} -46 q^{-64} +21 q^{-66} +2 q^{-68} -13 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: {10_123,}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {K11a87, K11a96,}

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{704}{3} -\frac{320}{3} -16 -\frac{256}{3} 128 -\frac{928}{3} -\frac{608}{3} \frac{1289}{15} \frac{388}{5} -\frac{5884}{45} \frac{391}{9} -\frac{871}{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 K11a28. 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        83  5
3       95   -4
1      118    3
-1     910     1
-3    710      -3
-5   59       4
-7  27        -5
-9 15         4
-11 2          -2
-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}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-4 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-3 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-2 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=-1 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=0 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{11}
r=1 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=2 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
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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K11a27.gif

K11a27

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K11a29