K11a35

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K11a34.gif

K11a34

K11a36.gif

K11a36

Contents

K11a35.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a35 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X8493 X14,5,15,6 X2837 X16,9,17,10 X18,12,19,11 X22,13,1,14 X6,15,7,16 X20,18,21,17 X12,20,13,19 X10,21,11,22
Gauss code 1, -4, 2, -1, 3, -8, 4, -2, 5, -11, 6, -10, 7, -3, 8, -5, 9, -6, 10, -9, 11, -7
Dowker-Thistlethwaite code 4 8 14 2 16 18 22 6 20 12 10
A Braid Representative
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A Morse Link Presentation K11a35 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number 1
3-genus 4
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a35/ThurstonBennequinNumber
Hyperbolic Volume 15.3717
A-Polynomial See Data:K11a35/A-polynomial

[edit Notes for K11a35'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 K11a35's four dimensional invariants]

Polynomial invariants

Alexander polynomial t^4-5 t^3+14 t^2-25 t+31-25 t^{-1} +14 t^{-2} -5 t^{-3} + t^{-4}
Conway polynomial z^8+3 z^6+4 z^4+2 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 121, 0 }
Jones polynomial q^6-4 q^5+8 q^4-13 q^3+17 q^2-19 q+20-16 q^{-1} +12 q^{-2} -7 q^{-3} +3 q^{-4} - q^{-5}
HOMFLY-PT polynomial (db, data sources) z^8-a^2 z^6-2 z^6 a^{-2} +6 z^6-4 a^2 z^4-8 z^4 a^{-2} +z^4 a^{-4} +15 z^4-6 a^2 z^2-11 z^2 a^{-2} +2 z^2 a^{-4} +17 z^2-3 a^2-5 a^{-2} + a^{-4} +8
Kauffman polynomial (db, data sources) z^{10} a^{-2} +z^{10}+4 a z^9+8 z^9 a^{-1} +4 z^9 a^{-3} +6 a^2 z^8+14 z^8 a^{-2} +6 z^8 a^{-4} +14 z^8+5 a^3 z^7+2 a z^7-5 z^7 a^{-1} +2 z^7 a^{-3} +4 z^7 a^{-5} +3 a^4 z^6-9 a^2 z^6-41 z^6 a^{-2} -13 z^6 a^{-4} +z^6 a^{-6} -39 z^6+a^5 z^5-7 a^3 z^5-14 a z^5-22 z^5 a^{-1} -26 z^5 a^{-3} -10 z^5 a^{-5} -5 a^4 z^4+10 a^2 z^4+39 z^4 a^{-2} +6 z^4 a^{-4} -2 z^4 a^{-6} +46 z^4-2 a^5 z^3+3 a^3 z^3+17 a z^3+31 z^3 a^{-1} +26 z^3 a^{-3} +7 z^3 a^{-5} +2 a^4 z^2-9 a^2 z^2-19 z^2 a^{-2} -2 z^2 a^{-4} +z^2 a^{-6} -27 z^2+a^5 z-a^3 z-7 a z-11 z a^{-1} -8 z a^{-3} -2 z a^{-5} +3 a^2+5 a^{-2} + a^{-4} +8
The A2 invariant -q^{14}+q^{12}-3 q^{10}+q^8+q^6-2 q^4+6 q^2-1+4 q^{-2} -2 q^{-6} +2 q^{-8} -4 q^{-10} + q^{-12} - q^{-16} + q^{-18}
The G2 invariant q^{80}-2 q^{78}+5 q^{76}-8 q^{74}+9 q^{72}-8 q^{70}+q^{68}+13 q^{66}-29 q^{64}+47 q^{62}-59 q^{60}+52 q^{58}-29 q^{56}-18 q^{54}+82 q^{52}-145 q^{50}+189 q^{48}-190 q^{46}+123 q^{44}-q^{42}-162 q^{40}+313 q^{38}-397 q^{36}+369 q^{34}-220 q^{32}-26 q^{30}+284 q^{28}-458 q^{26}+482 q^{24}-334 q^{22}+71 q^{20}+198 q^{18}-370 q^{16}+362 q^{14}-175 q^{12}-93 q^{10}+339 q^8-419 q^6+293 q^4+9 q^2-356+618 q^{-2} -666 q^{-4} +474 q^{-6} -91 q^{-8} -335 q^{-10} +668 q^{-12} -772 q^{-14} +624 q^{-16} -278 q^{-18} -136 q^{-20} +452 q^{-22} -573 q^{-24} +464 q^{-26} -186 q^{-28} -133 q^{-30} +350 q^{-32} -382 q^{-34} +211 q^{-36} +72 q^{-38} -347 q^{-40} +480 q^{-42} -419 q^{-44} +178 q^{-46} +137 q^{-48} -410 q^{-50} +541 q^{-52} -482 q^{-54} +277 q^{-56} -11 q^{-58} -226 q^{-60} +352 q^{-62} -351 q^{-64} +255 q^{-66} -106 q^{-68} -28 q^{-70} +113 q^{-72} -139 q^{-74} +116 q^{-76} -69 q^{-78} +26 q^{-80} +7 q^{-82} -22 q^{-84} +21 q^{-86} -16 q^{-88} +8 q^{-90} -3 q^{-92} + q^{-94}

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a316,}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {K11a36, K11a316,}

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{28}{3} -\frac{52}{3} -64 -\frac{176}{3} -\frac{32}{3} 24 \frac{256}{3} 32 \frac{224}{3} -\frac{416}{3} \frac{1231}{15} \frac{1516}{15} -\frac{7916}{45} \frac{17}{9} -\frac{929}{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 K11a35. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-5-4-3-2-10123456χ
13           11
11          3 -3
9         51 4
7        83  -5
5       95   4
3      108    -2
1     109     1
-1    711      4
-3   59       -4
-5  27        5
-7 15         -4
-9 2          2
-111           -1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-1 i=1
r=-5 {\mathbb Z}
r=-4 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-3 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-2 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-1 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=0 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{10}
r=1 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=2 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=3 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=4 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=5 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=6 {\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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K11a34.gif

K11a34

K11a36.gif

K11a36