K11a47

From Knot Atlas
Jump to: navigation, search

K11a46.gif

K11a46

K11a48.gif

K11a48

Contents

K11a47.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a47 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X8493 X14,5,15,6 X2837 X20,9,21,10 X22,11,1,12 X18,14,19,13 X6,15,7,16 X12,18,13,17 X16,20,17,19 X10,21,11,22
Gauss code 1, -4, 2, -1, 3, -8, 4, -2, 5, -11, 6, -9, 7, -3, 8, -10, 9, -7, 10, -5, 11, -6
Dowker-Thistlethwaite code 4 8 14 2 20 22 18 6 12 16 10
A Braid Representative
BraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart3.gifBraidPart3.gifBraidPart2.gifBraidPart0.gifBraidPart3.gifBraidPart3.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart4.gifBraidPart4.gifBraidPart1.gifBraidPart1.gifBraidPart4.gifBraidPart4.gifBraidPart1.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart2.gif
A Morse Link Presentation K11a47 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number \{1,2\}
3-genus 4
Bridge index 4
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a47/ThurstonBennequinNumber
Hyperbolic Volume 15.5977
A-Polynomial See Data:K11a47/A-polynomial

[edit Notes for K11a47's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus [2,4]
Rasmussen s-Invariant 0

[edit Notes for K11a47's four dimensional invariants]

Polynomial invariants

Alexander polynomial t^4-5 t^3+14 t^2-24 t+29-24 t^{-1} +14 t^{-2} -5 t^{-3} + t^{-4}
Conway polynomial z^8+3 z^6+4 z^4+3 z^2+1
2nd Alexander ideal (db, data sources) \left\{t^2-t+1\right\}
Determinant and Signature { 117, 0 }
Jones polynomial q^6-3 q^5+6 q^4-12 q^3+16 q^2-18 q+20-16 q^{-1} +13 q^{-2} -8 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^4-9 z^4 a^{-2} +z^4 a^{-4} +16 z^4-7 a^2 z^2-15 z^2 a^{-2} +3 z^2 a^{-4} +22 z^2-5 a^2-9 a^{-2} +2 a^{-4} +13
Kauffman polynomial (db, data sources) z^{10} a^{-2} +z^{10}+4 a z^9+7 z^9 a^{-1} +3 z^9 a^{-3} +7 a^2 z^8+10 z^8 a^{-2} +4 z^8 a^{-4} +13 z^8+6 a^3 z^7+5 a z^7-2 z^7 a^{-1} +2 z^7 a^{-3} +3 z^7 a^{-5} +3 a^4 z^6-11 a^2 z^6-28 z^6 a^{-2} -8 z^6 a^{-4} +z^6 a^{-6} -33 z^6+a^5 z^5-10 a^3 z^5-25 a z^5-27 z^5 a^{-1} -22 z^5 a^{-3} -9 z^5 a^{-5} -4 a^4 z^4+11 a^2 z^4+28 z^4 a^{-2} +3 z^4 a^{-4} -3 z^4 a^{-6} +37 z^4-2 a^5 z^3+8 a^3 z^3+31 a z^3+42 z^3 a^{-1} +30 z^3 a^{-3} +9 z^3 a^{-5} +a^4 z^2-11 a^2 z^2-19 z^2 a^{-2} -z^2 a^{-4} +2 z^2 a^{-6} -28 z^2+a^5 z-4 a^3 z-15 a z-21 z a^{-1} -15 z a^{-3} -4 z a^{-5} +5 a^2+9 a^{-2} +2 a^{-4} +13
The A2 invariant -q^{14}+q^{12}-4 q^{10}-q^4+8 q^2+1+6 q^{-2} - q^{-4} -3 q^{-6} -5 q^{-10} + q^{-12} + q^{-18}
The G2 invariant q^{80}-2 q^{78}+5 q^{76}-8 q^{74}+10 q^{72}-10 q^{70}+4 q^{68}+11 q^{66}-32 q^{64}+56 q^{62}-76 q^{60}+72 q^{58}-44 q^{56}-17 q^{54}+110 q^{52}-199 q^{50}+259 q^{48}-246 q^{46}+131 q^{44}+52 q^{42}-273 q^{40}+435 q^{38}-479 q^{36}+360 q^{34}-112 q^{32}-197 q^{30}+437 q^{28}-512 q^{26}+392 q^{24}-123 q^{22}-181 q^{20}+367 q^{18}-368 q^{16}+183 q^{14}+119 q^{12}-385 q^{10}+511 q^8-392 q^6+97 q^4+299 q^2-617+749 q^{-2} -608 q^{-4} +270 q^{-6} +167 q^{-8} -537 q^{-10} +735 q^{-12} -661 q^{-14} +376 q^{-16} +9 q^{-18} -349 q^{-20} +498 q^{-22} -427 q^{-24} +162 q^{-26} +140 q^{-28} -358 q^{-30} +388 q^{-32} -232 q^{-34} -64 q^{-36} +347 q^{-38} -505 q^{-40} +463 q^{-42} -264 q^{-44} -42 q^{-46} +310 q^{-48} -453 q^{-50} +451 q^{-52} -307 q^{-54} +106 q^{-56} +93 q^{-58} -223 q^{-60} +258 q^{-62} -217 q^{-64} +131 q^{-66} -34 q^{-68} -36 q^{-70} +74 q^{-72} -78 q^{-74} +63 q^{-76} -35 q^{-78} +13 q^{-80} +3 q^{-82} -12 q^{-84} +10 q^{-86} -9 q^{-88} +5 q^{-90} -2 q^{-92} + q^{-94}

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a44, K11a109,}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {K11a44,}

Vassiliev invariants

V2 and V3: (3, -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
12 -16 72 62 2 -192 -\frac{832}{3} -\frac{64}{3} -48 288 128 744 24 \frac{8751}{10} -\frac{106}{15} \frac{1514}{5} \frac{59}{2} \frac{271}{10}

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 K11a47. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-5-4-3-2-10123456χ
13           11
11          2 -2
9         41 3
7        82  -6
5       84   4
3      108    -2
1     108     2
-1    711      4
-3   69       -3
-5  27        5
-7 16         -5
-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}^{6}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-2 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
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}^{8}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=2 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=3 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=4 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=5 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
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

Read me first: Modifying Knot Pages.

See/edit the Hoste-Thistlethwaite Knot Page master template (intermediate).

See/edit the Hoste-Thistlethwaite_Splice_Base (expert).

Back to the top.

K11a46.gif

K11a46

K11a48.gif

K11a48