K11a129

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K11a128.gif

K11a128

K11a130.gif

K11a130

Contents

K11a129.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a129 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X10,3,11,4 X14,6,15,5 X20,7,21,8 X16,9,17,10 X2,11,3,12 X18,13,19,14 X22,16,1,15 X12,17,13,18 X6,19,7,20 X8,21,9,22
Gauss code 1, -6, 2, -1, 3, -10, 4, -11, 5, -2, 6, -9, 7, -3, 8, -5, 9, -7, 10, -4, 11, -8
Dowker-Thistlethwaite code 4 10 14 20 16 2 18 22 12 6 8
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
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BraidPart0.gifBraidPart4.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart4.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gif
A Morse Link Presentation K11a129 ML.gif

Three dimensional invariants

Symmetry type Chiral
Unknotting number 2
3-genus 4
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a129/ThurstonBennequinNumber
Hyperbolic Volume 15.5089
A-Polynomial See Data:K11a129/A-polynomial

[edit Notes for K11a129's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11a129's four dimensional invariants]

Polynomial invariants

Alexander polynomial t^4-6 t^3+15 t^2-22 t+25-22 t^{-1} +15 t^{-2} -6 t^{-3} + t^{-4}
Conway polynomial z^8+2 z^6-z^4+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 113, -4 }
Jones polynomial -q+4-7 q^{-1} +12 q^{-2} -15 q^{-3} +18 q^{-4} -18 q^{-5} +15 q^{-6} -12 q^{-7} +7 q^{-8} -3 q^{-9} + q^{-10}
HOMFLY-PT polynomial (db, data sources) z^4 a^8+3 z^2 a^8+2 a^8-2 z^6 a^6-8 z^4 a^6-10 z^2 a^6-5 a^6+z^8 a^4+5 z^6 a^4+9 z^4 a^4+8 z^2 a^4+3 a^4-z^6 a^2-3 z^4 a^2-z^2 a^2+a^2
Kauffman polynomial (db, data sources) z^4 a^{12}-z^2 a^{12}+3 z^5 a^{11}-2 z^3 a^{11}+6 z^6 a^{10}-6 z^4 a^{10}+3 z^2 a^{10}+9 z^7 a^9-15 z^5 a^9+14 z^3 a^9-5 z a^9+9 z^8 a^8-15 z^6 a^8+10 z^4 a^8-4 z^2 a^8+2 a^8+6 z^9 a^7-4 z^7 a^7-15 z^5 a^7+20 z^3 a^7-9 z a^7+2 z^{10} a^6+10 z^8 a^6-41 z^6 a^6+43 z^4 a^6-22 z^2 a^6+5 a^6+11 z^9 a^5-30 z^7 a^5+18 z^5 a^5+z^3 a^5-3 z a^5+2 z^{10} a^4+5 z^8 a^4-35 z^6 a^4+42 z^4 a^4-18 z^2 a^4+3 a^4+5 z^9 a^3-16 z^7 a^3+12 z^5 a^3-z^3 a^3+z a^3+4 z^8 a^2-15 z^6 a^2+16 z^4 a^2-4 z^2 a^2-a^2+z^7 a-3 z^5 a+2 z^3 a
The A2 invariant q^{30}+2 q^{24}-3 q^{22}+q^{20}-3 q^{18}-2 q^{16}+2 q^{14}-3 q^{12}+5 q^{10}-q^8+2 q^6+2 q^4-q^2+2- q^{-2}
The G2 invariant q^{162}-2 q^{160}+4 q^{158}-6 q^{156}+6 q^{154}-5 q^{152}+9 q^{148}-19 q^{146}+29 q^{144}-37 q^{142}+33 q^{140}-20 q^{138}-6 q^{136}+46 q^{134}-80 q^{132}+105 q^{130}-107 q^{128}+75 q^{126}-22 q^{124}-56 q^{122}+137 q^{120}-193 q^{118}+212 q^{116}-171 q^{114}+77 q^{112}+61 q^{110}-193 q^{108}+289 q^{106}-295 q^{104}+195 q^{102}-23 q^{100}-162 q^{98}+283 q^{96}-276 q^{94}+149 q^{92}+57 q^{90}-242 q^{88}+303 q^{86}-219 q^{84}-6 q^{82}+257 q^{80}-431 q^{78}+431 q^{76}-257 q^{74}-33 q^{72}+325 q^{70}-516 q^{68}+523 q^{66}-362 q^{64}+81 q^{62}+206 q^{60}-404 q^{58}+453 q^{56}-328 q^{54}+106 q^{52}+135 q^{50}-300 q^{48}+319 q^{46}-185 q^{44}-38 q^{42}+255 q^{40}-358 q^{38}+300 q^{36}-96 q^{34}-158 q^{32}+364 q^{30}-427 q^{28}+334 q^{26}-126 q^{24}-111 q^{22}+284 q^{20}-335 q^{18}+273 q^{16}-133 q^{14}-9 q^{12}+107 q^{10}-144 q^8+125 q^6-74 q^4+26 q^2+11-26 q^{-2} +23 q^{-4} -17 q^{-6} +8 q^{-8} -3 q^{-10} + q^{-12}

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

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

Vassiliev invariants

V2 and V3: (0, 3)
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
0 24 0 -96 8 0 144 -64 -40 0 288 0 0 608 -\frac{136}{3} \frac{2296}{3} \frac{64}{3} 80

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=-4 is the signature of K11a129. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-8-7-6-5-4-3-2-10123χ
3           1-1
1          3 3
-1         41 -3
-3        83  5
-5       85   -3
-7      107    3
-9     88     0
-11    710      -3
-13   58       3
-15  27        -5
-17 15         4
-19 2          -2
-211           1
Integral Khovanov Homology

(db, data source)

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

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K11a128

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K11a130