From Knot Atlas
Jump to: navigation, search






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

Visit K11a157 at Knotilus!

Knot presentations

Planar diagram presentation X4251 X10,3,11,4 X18,5,19,6 X12,8,13,7 X16,10,17,9 X2,11,3,12 X20,13,21,14 X8,16,9,15 X22,17,1,18 X6,19,7,20 X14,21,15,22
Gauss code 1, -6, 2, -1, 3, -10, 4, -8, 5, -2, 6, -4, 7, -11, 8, -5, 9, -3, 10, -7, 11, -9
Dowker-Thistlethwaite code 4 10 18 12 16 2 20 8 22 6 14
A Braid Representative
A Morse Link Presentation K11a157 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:K11a157/ThurstonBennequinNumber
Hyperbolic Volume 16.8128
A-Polynomial See Data:K11a157/A-polynomial

[edit Notes for K11a157's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11a157's four dimensional invariants]

Polynomial invariants

Alexander polynomial -t^4+6 t^3-16 t^2+28 t-33+28 t^{-1} -16 t^{-2} +6 t^{-3} - t^{-4}
Conway polynomial -z^8-2 z^6+2 z^2+1
2nd Alexander ideal (db, data sources) \left\{2,t^4+t^2+1\right\}
Determinant and Signature { 135, -2 }
Jones polynomial q^3-4 q^2+9 q-14+19 q^{-1} -22 q^{-2} +22 q^{-3} -18 q^{-4} +14 q^{-5} -8 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^4-10 a^2 z^4+3 z^4-3 a^6 z^2+12 a^4 z^2-10 a^2 z^2+3 z^2-3 a^6+7 a^4-5 a^2+2
Kauffman polynomial (db, data sources) 2 a^4 z^{10}+2 a^2 z^{10}+6 a^5 z^9+12 a^3 z^9+6 a z^9+8 a^6 z^8+13 a^4 z^8+12 a^2 z^8+7 z^8+6 a^7 z^7-3 a^5 z^7-20 a^3 z^7-7 a z^7+4 z^7 a^{-1} +3 a^8 z^6-14 a^6 z^6-39 a^4 z^6-39 a^2 z^6+z^6 a^{-2} -16 z^6+a^9 z^5-9 a^7 z^5-8 a^5 z^5+4 a^3 z^5-7 a z^5-9 z^5 a^{-1} -4 a^8 z^4+16 a^6 z^4+46 a^4 z^4+38 a^2 z^4-2 z^4 a^{-2} +10 z^4-2 a^9 z^3+6 a^7 z^3+13 a^5 z^3+8 a^3 z^3+8 a z^3+5 z^3 a^{-1} +a^8 z^2-12 a^6 z^2-28 a^4 z^2-20 a^2 z^2+z^2 a^{-2} -4 z^2+a^9 z-2 a^7 z-5 a^5 z-3 a^3 z-2 a z-z a^{-1} +3 a^6+7 a^4+5 a^2+2
The A2 invariant -q^{24}-q^{20}-3 q^{18}+4 q^{16}-q^{14}+4 q^{12}+3 q^{10}-3 q^8+3 q^6-6 q^4+3 q^2- q^{-2} +3 q^{-4} -2 q^{-6} + q^{-8}
The G2 invariant q^{128}-2 q^{126}+5 q^{124}-8 q^{122}+10 q^{120}-10 q^{118}+4 q^{116}+10 q^{114}-30 q^{112}+53 q^{110}-73 q^{108}+74 q^{106}-53 q^{104}+89 q^{100}-184 q^{98}+263 q^{96}-285 q^{94}+205 q^{92}-43 q^{90}-200 q^{88}+444 q^{86}-594 q^{84}+578 q^{82}-353 q^{80}-33 q^{78}+442 q^{76}-718 q^{74}+741 q^{72}-503 q^{70}+62 q^{68}+382 q^{66}-634 q^{64}+590 q^{62}-242 q^{60}-225 q^{58}+619 q^{56}-714 q^{54}+457 q^{52}+41 q^{50}-598 q^{48}+990 q^{46}-1019 q^{44}+687 q^{42}-80 q^{40}-565 q^{38}+1029 q^{36}-1139 q^{34}+855 q^{32}-321 q^{30}-291 q^{28}+734 q^{26}-857 q^{24}+643 q^{22}-176 q^{20}-321 q^{18}+622 q^{16}-611 q^{14}+275 q^{12}+201 q^{10}-617 q^8+783 q^6-619 q^4+209 q^2+290-671 q^{-2} +802 q^{-4} -649 q^{-6} +303 q^{-8} +78 q^{-10} -371 q^{-12} +490 q^{-14} -435 q^{-16} +279 q^{-18} -80 q^{-20} -74 q^{-22} +152 q^{-24} -163 q^{-26} +122 q^{-28} -66 q^{-30} +20 q^{-32} +12 q^{-34} -24 q^{-36} +22 q^{-38} -16 q^{-40} +8 q^{-42} -3 q^{-44} + q^{-46}

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a264, K11a305,}

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

Vassiliev invariants

V2 and V3: (2, -5)
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 -40 32 \frac{412}{3} \frac{44}{3} -320 -\frac{1840}{3} -\frac{160}{3} -104 \frac{256}{3} 800 \frac{3296}{3} \frac{352}{3} \frac{45871}{15} -\frac{108}{5} \frac{50644}{45} \frac{929}{9} \frac{1711}{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 K11a157. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
7           11
5          3 -3
3         61 5
1        83  -5
-1       116   5
-3      129    -3
-5     1010     0
-7    812      4
-9   610       -4
-11  28        6
-13 16         -5
-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}^{6}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-4 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=-3 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=-2 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=-1 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{12}
r=0 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{11}
r=1 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
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.