K11a3

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K11a2.gif

K11a2

K11a4.gif

K11a4

Contents

K11a3.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a3 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X8394 X10,6,11,5 X14,8,15,7 X2,9,3,10 X18,11,19,12 X6,14,7,13 X20,15,21,16 X22,17,1,18 X12,19,13,20 X16,21,17,22
Gauss code 1, -5, 2, -1, 3, -7, 4, -2, 5, -3, 6, -10, 7, -4, 8, -11, 9, -6, 10, -8, 11, -9
Dowker-Thistlethwaite code 4 8 10 14 2 18 6 20 22 12 16
A Braid Representative
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BraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
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A Morse Link Presentation K11a3 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:K11a3/ThurstonBennequinNumber
Hyperbolic Volume 15.55
A-Polynomial See Data:K11a3/A-polynomial

[edit Notes for K11a3'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 K11a3's four dimensional invariants]

Polynomial invariants

Alexander polynomial -t^4+5 t^3-13 t^2+24 t-29+24 t^{-1} -13 t^{-2} +5 t^{-3} - t^{-4}
Conway polynomial -z^8-3 z^6-3 z^4+z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 115, -2 }
Jones polynomial q^3-3 q^2+7 q-12+16 q^{-1} -18 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^6-6 a^2 z^6+z^6-a^6 z^4+9 a^4 z^4-15 a^2 z^4+4 z^4-3 a^6 z^2+15 a^4 z^2-17 a^2 z^2+6 z^2-3 a^6+8 a^4-7 a^2+3
Kauffman polynomial (db, data sources) a^4 z^{10}+a^2 z^{10}+4 a^5 z^9+8 a^3 z^9+4 a z^9+6 a^6 z^8+13 a^4 z^8+12 a^2 z^8+5 z^8+5 a^7 z^7+a^5 z^7-11 a^3 z^7-4 a z^7+3 z^7 a^{-1} +3 a^8 z^6-9 a^6 z^6-38 a^4 z^6-40 a^2 z^6+z^6 a^{-2} -13 z^6+a^9 z^5-7 a^7 z^5-11 a^5 z^5-2 a^3 z^5-7 a z^5-8 z^5 a^{-1} -5 a^8 z^4+9 a^6 z^4+48 a^4 z^4+49 a^2 z^4-3 z^4 a^{-2} +12 z^4-2 a^9 z^3+3 a^7 z^3+12 a^5 z^3+10 a^3 z^3+9 a z^3+6 z^3 a^{-1} +2 a^8 z^2-8 a^6 z^2-30 a^4 z^2-30 a^2 z^2+2 z^2 a^{-2} -8 z^2+a^9 z-a^7 z-4 a^5 z-4 a^3 z-4 a z-2 z a^{-1} +3 a^6+8 a^4+7 a^2+3
The A2 invariant -q^{24}-q^{20}-2 q^{18}+4 q^{16}-q^{14}+3 q^{12}+2 q^{10}-2 q^8+3 q^6-5 q^4+2 q^2-1- q^{-2} +3 q^{-4} - 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}-29 q^{112}+47 q^{110}-59 q^{108}+53 q^{106}-30 q^{104}-20 q^{102}+87 q^{100}-150 q^{98}+191 q^{96}-186 q^{94}+112 q^{92}+17 q^{90}-181 q^{88}+324 q^{86}-389 q^{84}+334 q^{82}-167 q^{80}-81 q^{78}+315 q^{76}-446 q^{74}+423 q^{72}-237 q^{70}-30 q^{68}+264 q^{66}-367 q^{64}+285 q^{62}-53 q^{60}-216 q^{58}+409 q^{56}-407 q^{54}+208 q^{52}+130 q^{50}-455 q^{48}+647 q^{46}-610 q^{44}+353 q^{42}+36 q^{40}-415 q^{38}+658 q^{36}-671 q^{34}+468 q^{32}-122 q^{30}-234 q^{28}+454 q^{26}-482 q^{24}+308 q^{22}-27 q^{20}-243 q^{18}+372 q^{16}-315 q^{14}+91 q^{12}+196 q^{10}-420 q^8+474 q^6-340 q^4+65 q^2+227-431 q^{-2} +485 q^{-4} -368 q^{-6} +159 q^{-8} +69 q^{-10} -235 q^{-12} +294 q^{-14} -251 q^{-16} +153 q^{-18} -39 q^{-20} -45 q^{-22} +88 q^{-24} -92 q^{-26} +69 q^{-28} -36 q^{-30} +11 q^{-32} +6 q^{-34} -13 q^{-36} +11 q^{-38} -9 q^{-40} +5 q^{-42} -2 q^{-44} + q^{-46}

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {K11a51, K11a331,}

Vassiliev invariants

V2 and V3: (1, -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
4 -32 8 \frac{350}{3} \frac{82}{3} -128 -\frac{992}{3} -\frac{128}{3} -32 \frac{32}{3} 512 \frac{1400}{3} \frac{328}{3} \frac{49231}{30} \frac{1618}{15} \frac{21422}{45} \frac{881}{18} \frac{1231}{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=-2 is the signature of K11a3. 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          2 -2
3         51 4
1        72  -5
-1       95   4
-3      108    -2
-5     98     1
-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}^{8}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=0 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{9}
r=1 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=2 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=3 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
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).

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K11a2.gif

K11a2

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K11a4