K11a15

K11a14

K11a16

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a15 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X8493 X12,5,13,6 X2837 X14,9,15,10 X18,12,19,11 X6,13,7,14 X20,15,21,16 X22,17,1,18 X10,20,11,19 X16,21,17,22
Gauss code	1, -4, 2, -1, 3, -7, 4, -2, 5, -10, 6, -3, 7, -5, 8, -11, 9, -6, 10, -8, 11, -9
Dowker-Thistlethwaite code	4 8 12 2 14 18 6 20 22 10 16

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Three dimensional invariants

Symmetry type Chiral Unknotting number [math]\displaystyle{ \{1,2\} }[/math] 3-genus 4 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a15/ThurstonBennequinNumber Hyperbolic Volume 15.1288 A-Polynomial See Data:K11a15/A-polynomial

[edit Notes for K11a15's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus [math]\displaystyle{ 4 }[/math] Rasmussen s-Invariant 2

[edit Notes for K11a15's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ -t^4+5 t^3-13 t^2+22 t-25+22 t^{-1} -13 t^{-2} +5 t^{-3} - t^{-4} }[/math]
Conway polynomial	[math]\displaystyle{ -z^8-3 z^6-3 z^4-z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 107, -2 }
Jones polynomial	[math]\displaystyle{ -q^4+3 q^3-6 q^2+11 q-14+17 q^{-1} -17 q^{-2} +15 q^{-3} -12 q^{-4} +7 q^{-5} -3 q^{-6} + q^{-7} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ -a^2 z^8+a^4 z^6-6 a^2 z^6+2 z^6+4 a^4 z^4-15 a^2 z^4-z^4 a^{-2} +9 z^4+6 a^4 z^2-18 a^2 z^2-3 z^2 a^{-2} +14 z^2+3 a^4-8 a^2-2 a^{-2} +8 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ a^2 z^{10}+z^{10}+5 a^3 z^9+8 a z^9+3 z^9 a^{-1} +9 a^4 z^8+15 a^2 z^8+3 z^8 a^{-2} +9 z^8+9 a^5 z^7+a^3 z^7-15 a z^7-6 z^7 a^{-1} +z^7 a^{-3} +6 a^6 z^6-16 a^4 z^6-52 a^2 z^6-12 z^6 a^{-2} -42 z^6+3 a^7 z^5-15 a^5 z^5-25 a^3 z^5-9 a z^5-6 z^5 a^{-1} -4 z^5 a^{-3} +a^8 z^4-6 a^6 z^4+13 a^4 z^4+58 a^2 z^4+16 z^4 a^{-2} +54 z^4-2 a^7 z^3+14 a^5 z^3+29 a^3 z^3+23 a z^3+15 z^3 a^{-1} +5 z^3 a^{-3} -a^8 z^2+3 a^6 z^2-7 a^4 z^2-33 a^2 z^2-9 z^2 a^{-2} -31 z^2-5 a^5 z-11 a^3 z-10 a z-6 z a^{-1} -2 z a^{-3} +3 a^4+8 a^2+2 a^{-2} +8 }[/math]
The A2 invariant	[math]\displaystyle{ q^{20}-q^{18}+3 q^{16}-q^{14}-q^{12}+q^{10}-5 q^8+2 q^6-3 q^4+2 q^2+3+4 q^{-4} - q^{-6} - q^{-12} }[/math]
The G2 invariant	[math]\displaystyle{ q^{114}-2 q^{112}+4 q^{110}-6 q^{108}+6 q^{106}-5 q^{104}+9 q^{100}-19 q^{98}+29 q^{96}-37 q^{94}+33 q^{92}-20 q^{90}-5 q^{88}+44 q^{86}-78 q^{84}+105 q^{82}-114 q^{80}+87 q^{78}-31 q^{76}-56 q^{74}+155 q^{72}-222 q^{70}+240 q^{68}-180 q^{66}+52 q^{64}+114 q^{62}-253 q^{60}+318 q^{58}-267 q^{56}+116 q^{54}+78 q^{52}-231 q^{50}+278 q^{48}-189 q^{46}+10 q^{44}+180 q^{42}-295 q^{40}+259 q^{38}-97 q^{36}-144 q^{34}+352 q^{32}-446 q^{30}+366 q^{28}-152 q^{26}-136 q^{24}+378 q^{22}-501 q^{20}+449 q^{18}-254 q^{16}-18 q^{14}+262 q^{12}-389 q^{10}+367 q^8-197 q^6-28 q^4+220 q^2-291+223 q^{-2} -32 q^{-4} -173 q^{-6} +318 q^{-8} -320 q^{-10} +188 q^{-12} +29 q^{-14} -235 q^{-16} +359 q^{-18} -346 q^{-20} +222 q^{-22} -39 q^{-24} -136 q^{-26} +236 q^{-28} -246 q^{-30} +181 q^{-32} -80 q^{-34} -17 q^{-36} +76 q^{-38} -97 q^{-40} +81 q^{-42} -48 q^{-44} +17 q^{-46} +5 q^{-48} -16 q^{-50} +14 q^{-52} -11 q^{-54} +6 q^{-56} -2 q^{-58} + q^{-60} }[/math]

Further Quantum Invariants

K11a15 Quantum Invariants.

Computer Talk

The above data is available with the Mathematica package KnotTheory`, as shown in the (simulated) Mathematica session below. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting. This Mathematica session is also available (albeit only for the knot 5_2) as the notebook PolynomialInvariantsSession.nb.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:=

AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of August 31, 2006, 11:25:27.5625. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["K11a15"];

In[4]:=

Alexander[K][t]

KnotTheory::loading: Loading precomputed data in PD4Knots`.

Out[4]=

[math]\displaystyle{ -t^4+5 t^3-13 t^2+22 t-25+22 t^{-1} -13 t^{-2} +5 t^{-3} - t^{-4} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ -z^8-3 z^6-3 z^4-z^2+1 }[/math]

In[6]:=

Alexander[K, 2][t]

KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.

Out[6]=

[math]\displaystyle{ \{1\} }[/math]

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 107, -2 }

In[8]:=

Jones[K][q]

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[8]=

[math]\displaystyle{ -q^4+3 q^3-6 q^2+11 q-14+17 q^{-1} -17 q^{-2} +15 q^{-3} -12 q^{-4} +7 q^{-5} -3 q^{-6} + q^{-7} }[/math]

In[9]:=

HOMFLYPT[K][a, z]

KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.

Out[9]=

[math]\displaystyle{ -a^2 z^8+a^4 z^6-6 a^2 z^6+2 z^6+4 a^4 z^4-15 a^2 z^4-z^4 a^{-2} +9 z^4+6 a^4 z^2-18 a^2 z^2-3 z^2 a^{-2} +14 z^2+3 a^4-8 a^2-2 a^{-2} +8 }[/math]

In[10]:=

Kauffman[K][a, z]

KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.

Out[10]=

[math]\displaystyle{ a^2 z^{10}+z^{10}+5 a^3 z^9+8 a z^9+3 z^9 a^{-1} +9 a^4 z^8+15 a^2 z^8+3 z^8 a^{-2} +9 z^8+9 a^5 z^7+a^3 z^7-15 a z^7-6 z^7 a^{-1} +z^7 a^{-3} +6 a^6 z^6-16 a^4 z^6-52 a^2 z^6-12 z^6 a^{-2} -42 z^6+3 a^7 z^5-15 a^5 z^5-25 a^3 z^5-9 a z^5-6 z^5 a^{-1} -4 z^5 a^{-3} +a^8 z^4-6 a^6 z^4+13 a^4 z^4+58 a^2 z^4+16 z^4 a^{-2} +54 z^4-2 a^7 z^3+14 a^5 z^3+29 a^3 z^3+23 a z^3+15 z^3 a^{-1} +5 z^3 a^{-3} -a^8 z^2+3 a^6 z^2-7 a^4 z^2-33 a^2 z^2-9 z^2 a^{-2} -31 z^2-5 a^5 z-11 a^3 z-10 a z-6 z a^{-1} -2 z a^{-3} +3 a^4+8 a^2+2 a^{-2} +8 }[/math]

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, [math]\displaystyle{ q\leftrightarrow q^{-1} }[/math]): {}

Computer Talk

The above data is available with the Mathematica package KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:=

AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of May 31, 2006, 14:15:20.091. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["K11a15"];

In[4]:=

{A = Alexander[K][t], J = Jones[K][q]}

KnotTheory::loading: Loading precomputed data in PD4Knots`.

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[4]=

{ [math]\displaystyle{ -t^4+5 t^3-13 t^2+22 t-25+22 t^{-1} -13 t^{-2} +5 t^{-3} - t^{-4} }[/math], [math]\displaystyle{ -q^4+3 q^3-6 q^2+11 q-14+17 q^{-1} -17 q^{-2} +15 q^{-3} -12 q^{-4} +7 q^{-5} -3 q^{-6} + q^{-7} }[/math] }

In[5]:=

DeleteCases[Select[AllKnots[], (A === Alexander[#][t]) &], K]

KnotTheory::loading: Loading precomputed data in DTCode4KnotsTo11`.

KnotTheory::credits: The GaussCode to PD conversion was written by Siddarth Sankaran at the University of Toronto in the summer of 2005.

Out[5]=

{}

In[6]:=

DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]

KnotTheory::loading: Loading precomputed data in Jones4Knots11`.

Out[6]=

{}

Vassiliev invariants

V2 and V3:

(-1, 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 [math]\displaystyle{ -4 }[/math] [math]\displaystyle{ 24 }[/math] [math]\displaystyle{ 8 }[/math] [math]\displaystyle{ \frac{34}{3} }[/math] [math]\displaystyle{ \frac{86}{3} }[/math] [math]\displaystyle{ -96 }[/math] [math]\displaystyle{ -272 }[/math] [math]\displaystyle{ -96 }[/math] [math]\displaystyle{ -104 }[/math] [math]\displaystyle{ -\frac{32}{3} }[/math] [math]\displaystyle{ 288 }[/math] [math]\displaystyle{ -\frac{136}{3} }[/math] [math]\displaystyle{ -\frac{344}{3} }[/math] [math]\displaystyle{ \frac{22289}{30} }[/math] [math]\displaystyle{ \frac{3622}{15} }[/math] [math]\displaystyle{ \frac{778}{45} }[/math] [math]\displaystyle{ \frac{2383}{18} }[/math] [math]\displaystyle{ -\frac{2671}{30} }[/math]

V_2,1 through V_6,9 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 [math]\displaystyle{ t^rq^j }[/math] are shown, along with their alternating sums [math]\displaystyle{ \chi }[/math] (fixed [math]\displaystyle{ j }[/math], alternation over [math]\displaystyle{ r }[/math]). The squares with yellow highlighting are those on the "critical diagonals", where [math]\displaystyle{ j-2r=s+1 }[/math] or [math]\displaystyle{ j-2r=s-1 }[/math], where [math]\displaystyle{ s= }[/math]-2 is the signature of K11a15. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 9                       1 -1 7                     2   2 5                   4 1   -3 3                 7 2     5 1               7 4       -3 -1             10 7         3 -3           8 8           0 -5         7 9             -2 -7       5 8               3 -9     2 7                 -5 -11   1 5                   4 -13   2                     -2 -15 1                       1

Integral Khovanov Homology (db, data source)

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

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.

K11a14 K11a16