From Knot Atlas
Jump to: navigation, search






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

Visit K11a113 at Knotilus!

Knot presentations

Planar diagram presentation X4251 X10,3,11,4 X14,6,15,5 X16,7,17,8 X18,9,19,10 X2,11,3,12 X20,13,21,14 X22,16,1,15 X8,17,9,18 X12,19,13,20 X6,21,7,22
Gauss code 1, -6, 2, -1, 3, -11, 4, -9, 5, -2, 6, -10, 7, -3, 8, -4, 9, -5, 10, -7, 11, -8
Dowker-Thistlethwaite code 4 10 14 16 18 2 20 22 8 12 6
A Braid Representative
A Morse Link Presentation K11a113 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:K11a113/ThurstonBennequinNumber
Hyperbolic Volume 15.6778
A-Polynomial See Data:K11a113/A-polynomial

[edit Notes for K11a113'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 K11a113's four dimensional invariants]

Polynomial invariants

Alexander polynomial t^4-6 t^3+15 t^2-21 t+23-21 t^{-1} +15 t^{-2} -6 t^{-3} + t^{-4}
Conway polynomial z^8+2 z^6-z^4+z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 109, -4 }
Jones polynomial -q+4-7 q^{-1} +12 q^{-2} -15 q^{-3} +17 q^{-4} -17 q^{-5} +15 q^{-6} -11 q^{-7} +6 q^{-8} -3 q^{-9} + q^{-10}
HOMFLY-PT polynomial (db, data sources) z^4 a^8+3 z^2 a^8+a^8-2 z^6 a^6-8 z^4 a^6-9 z^2 a^6-3 a^6+z^8 a^4+5 z^6 a^4+9 z^4 a^4+8 z^2 a^4+2 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}-3 z^3 a^{11}+z a^{11}+5 z^6 a^{10}-4 z^4 a^{10}+z^2 a^{10}+7 z^7 a^9-8 z^5 a^9+3 z^3 a^9+z a^9+8 z^8 a^8-14 z^6 a^8+11 z^4 a^8-4 z^2 a^8+a^8+6 z^9 a^7-8 z^7 a^7-2 z^5 a^7+4 z^3 a^7-z a^7+2 z^{10} a^6+9 z^8 a^6-40 z^6 a^6+43 z^4 a^6-19 z^2 a^6+3 a^6+11 z^9 a^5-32 z^7 a^5+24 z^5 a^5-6 z^3 a^5+2 z^{10} a^4+5 z^8 a^4-36 z^6 a^4+43 z^4 a^4-17 z^2 a^4+2 a^4+5 z^9 a^3-16 z^7 a^3+12 z^5 a^3-2 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}-q^{26}+q^{24}-3 q^{22}+2 q^{20}-q^{18}-q^{16}+2 q^{14}-4 q^{12}+4 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}+5 q^{154}-4 q^{152}-2 q^{150}+10 q^{148}-18 q^{146}+26 q^{144}-29 q^{142}+24 q^{140}-9 q^{138}-12 q^{136}+39 q^{134}-62 q^{132}+75 q^{130}-75 q^{128}+50 q^{126}-10 q^{124}-37 q^{122}+93 q^{120}-130 q^{118}+154 q^{116}-145 q^{114}+86 q^{112}+10 q^{110}-131 q^{108}+232 q^{106}-267 q^{104}+218 q^{102}-83 q^{100}-96 q^{98}+246 q^{96}-295 q^{94}+211 q^{92}-33 q^{90}-171 q^{88}+281 q^{86}-247 q^{84}+79 q^{82}+156 q^{80}-341 q^{78}+390 q^{76}-277 q^{74}+32 q^{72}+239 q^{70}-441 q^{68}+487 q^{66}-364 q^{64}+129 q^{62}+143 q^{60}-350 q^{58}+432 q^{56}-363 q^{54}+171 q^{52}+63 q^{50}-261 q^{48}+328 q^{46}-237 q^{44}+42 q^{42}+182 q^{40}-318 q^{38}+305 q^{36}-145 q^{34}-96 q^{32}+312 q^{30}-404 q^{28}+340 q^{26}-146 q^{24}-86 q^{22}+266 q^{20}-326 q^{18}+274 q^{16}-141 q^{14}-3 q^{12}+105 q^{10}-145 q^8+126 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: (1, 0)
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
4 0 8 -\frac{34}{3} \frac{34}{3} 0 0 -64 -32 \frac{32}{3} 0 -\frac{136}{3} \frac{136}{3} \frac{5071}{30} -\frac{1742}{15} \frac{25982}{45} -\frac{175}{18} \frac{2671}{30}

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 K11a113. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
3           1-1
1          3 3
-1         41 -3
-3        83  5
-5       85   -3
-7      97    2
-9     88     0
-11    79      -2
-13   48       4
-15  27        -5
-17 14         3
-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}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-5 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-4 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=-3 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=-2 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
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).

See/edit the Hoste-Thistlethwaite_Splice_Base (expert).

Back to the top.