K11a66

From Knot Atlas
Jump to: navigation, search

K11a65.gif

K11a65

K11a67.gif

K11a67

Contents

K11a66.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a66 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X8394 X16,5,17,6 X12,8,13,7 X2,9,3,10 X18,12,19,11 X20,13,21,14 X22,15,1,16 X10,18,11,17 X6,19,7,20 X14,21,15,22
Gauss code 1, -5, 2, -1, 3, -10, 4, -2, 5, -9, 6, -4, 7, -11, 8, -3, 9, -6, 10, -7, 11, -8
Dowker-Thistlethwaite code 4 8 16 12 2 18 20 22 10 6 14
A Braid Representative
BraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart3.gifBraidPart3.gif
BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart4.gifBraidPart4.gif
A Morse Link Presentation K11a66 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:K11a66/ThurstonBennequinNumber
Hyperbolic Volume 15.8011
A-Polynomial See Data:K11a66/A-polynomial

[edit Notes for K11a66's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 4
Rasmussen s-Invariant 2

[edit Notes for K11a66's four dimensional invariants]

Polynomial invariants

Alexander polynomial -t^4+6 t^3-15 t^2+24 t-27+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 { 119, -2 }
Jones polynomial q^3-4 q^2+8 q-12+17 q^{-1} -19 q^{-2} +19 q^{-3} -16 q^{-4} +12 q^{-5} -7 q^{-6} +3 q^{-7} - q^{-8}
HOMFLY-PT polynomial (db, data sources) -a^2 z^8+2 a^4 z^6-5 a^2 z^6+z^6-a^6 z^4+8 a^4 z^4-9 a^2 z^4+3 z^4-3 a^6 z^2+10 a^4 z^2-7 a^2 z^2+2 z^2-2 a^6+4 a^4-2 a^2+1
Kauffman polynomial (db, data sources) 2 a^4 z^{10}+2 a^2 z^{10}+5 a^5 z^9+11 a^3 z^9+6 a z^9+6 a^6 z^8+7 a^4 z^8+8 a^2 z^8+7 z^8+5 a^7 z^7-6 a^5 z^7-28 a^3 z^7-13 a z^7+4 z^7 a^{-1} +3 a^8 z^6-9 a^6 z^6-26 a^4 z^6-35 a^2 z^6+z^6 a^{-2} -20 z^6+a^9 z^5-7 a^7 z^5+5 a^5 z^5+29 a^3 z^5+6 a z^5-10 z^5 a^{-1} -5 a^8 z^4+8 a^6 z^4+36 a^4 z^4+41 a^2 z^4-2 z^4 a^{-2} +16 z^4-2 a^9 z^3+2 a^7 z^3-3 a^5 z^3-12 a^3 z^3-a z^3+4 z^3 a^{-1} +2 a^8 z^2-7 a^6 z^2-21 a^4 z^2-17 a^2 z^2-5 z^2+a^9 z+2 a^3 z+a z+2 a^6+4 a^4+2 a^2+1
The A2 invariant -q^{24}-2 q^{18}+3 q^{16}-2 q^{14}+2 q^{12}+2 q^{10}-2 q^8+4 q^6-4 q^4+3 q^2- q^{-2} +2 q^{-4} -2 q^{-6} + q^{-8}
The G2 invariant q^{128}-2 q^{126}+5 q^{124}-8 q^{122}+9 q^{120}-8 q^{118}+q^{116}+13 q^{114}-30 q^{112}+48 q^{110}-58 q^{108}+49 q^{106}-22 q^{104}-26 q^{102}+86 q^{100}-138 q^{98}+166 q^{96}-158 q^{94}+93 q^{92}+8 q^{90}-140 q^{88}+268 q^{86}-338 q^{84}+321 q^{82}-195 q^{80}-17 q^{78}+243 q^{76}-408 q^{74}+439 q^{72}-312 q^{70}+66 q^{68}+200 q^{66}-367 q^{64}+359 q^{62}-166 q^{60}-109 q^{58}+341 q^{56}-418 q^{54}+280 q^{52}+13 q^{50}-346 q^{48}+586 q^{46}-611 q^{44}+417 q^{42}-53 q^{40}-338 q^{38}+616 q^{36}-692 q^{34}+539 q^{32}-221 q^{30}-147 q^{28}+437 q^{26}-537 q^{24}+432 q^{22}-158 q^{20}-154 q^{18}+360 q^{16}-388 q^{14}+209 q^{12}+79 q^{10}-342 q^8+475 q^6-401 q^4+155 q^2+161-417 q^{-2} +514 q^{-4} -431 q^{-6} +212 q^{-8} +43 q^{-10} -244 q^{-12} +335 q^{-14} -307 q^{-16} +202 q^{-18} -63 q^{-20} -51 q^{-22} +109 q^{-24} -122 q^{-26} +95 q^{-28} -53 q^{-30} +19 q^{-32} +9 q^{-34} -19 q^{-36} +18 q^{-38} -14 q^{-40} +7 q^{-42} -3 q^{-44} + q^{-46}

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a163,}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {K11a121,}

Vassiliev invariants

V2 and V3: (2, -4)
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 -32 32 \frac{316}{3} \frac{20}{3} -256 -\frac{1376}{3} -\frac{32}{3} -96 \frac{256}{3} 512 \frac{2528}{3} \frac{160}{3} \frac{30991}{15} -\frac{764}{15} \frac{30604}{45} \frac{1025}{9} \frac{991}{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=-2 is the signature of K11a66. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-7-6-5-4-3-2-101234χ
7           11
5          3 -3
3         51 4
1        73  -4
-1       105   5
-3      108    -2
-5     99     0
-7    710      3
-9   59       -4
-11  27        5
-13 15         -4
-15 2          2
-171           -1
Integral Khovanov Homology

(db, data source)

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

K11a65.gif

K11a65

K11a67.gif

K11a67