K11a158

K11a157

K11a159

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a158 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X10,4,11,3 X18,5,19,6 X14,7,15,8 X2,10,3,9 X16,11,17,12 X20,14,21,13 X8,15,9,16 X22,17,1,18 X6,19,7,20 X12,22,13,21
Gauss code	1, -5, 2, -1, 3, -10, 4, -8, 5, -2, 6, -11, 7, -4, 8, -6, 9, -3, 10, -7, 11, -9
Dowker-Thistlethwaite code	4 10 18 14 2 16 20 8 22 6 12

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

Three dimensional invariants

Symmetry type Reversible 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:K11a158/ThurstonBennequinNumber Hyperbolic Volume 16.235 A-Polynomial See Data:K11a158/A-polynomial

[edit Notes for K11a158'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 K11a158's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ -t^4+5 t^3-14 t^2+25 t-29+25 t^{-1} -14 t^{-2} +5 t^{-3} - t^{-4} }[/math]
Conway polynomial	[math]\displaystyle{ -z^8-3 z^6-4 z^4-2 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 119, -2 }
Jones polynomial	[math]\displaystyle{ -q^4+4 q^3-8 q^2+13 q-16+19 q^{-1} -19 q^{-2} +16 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} +8 z^4+6 a^4 z^2-17 a^2 z^2-2 z^2 a^{-2} +11 z^2+3 a^4-7 a^2- a^{-2} +6 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ 2 a^2 z^{10}+2 z^{10}+7 a^3 z^9+12 a z^9+5 z^9 a^{-1} +10 a^4 z^8+14 a^2 z^8+4 z^8 a^{-2} +8 z^8+9 a^5 z^7-7 a^3 z^7-30 a z^7-13 z^7 a^{-1} +z^7 a^{-3} +6 a^6 z^6-19 a^4 z^6-57 a^2 z^6-14 z^6 a^{-2} -46 z^6+3 a^7 z^5-14 a^5 z^5-12 a^3 z^5+10 a z^5+2 z^5 a^{-1} -3 z^5 a^{-3} +a^8 z^4-6 a^6 z^4+17 a^4 z^4+65 a^2 z^4+15 z^4 a^{-2} +56 z^4-2 a^7 z^3+12 a^5 z^3+18 a^3 z^3+10 a z^3+9 z^3 a^{-1} +3 z^3 a^{-3} -a^8 z^2+3 a^6 z^2-9 a^4 z^2-34 a^2 z^2-6 z^2 a^{-2} -27 z^2-4 a^5 z-7 a^3 z-5 a z-3 z a^{-1} -z a^{-3} +3 a^4+7 a^2+ a^{-2} +6 }[/math]
The A2 invariant	[math]\displaystyle{ q^{20}-q^{18}+3 q^{16}-q^{14}-q^{12}+2 q^{10}-5 q^8+2 q^6-3 q^4+q^2+3- q^{-2} +4 q^{-4} - q^{-6} + q^{-10} - 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}-36 q^{94}+31 q^{92}-18 q^{90}-6 q^{88}+41 q^{86}-71 q^{84}+95 q^{82}-101 q^{80}+80 q^{78}-38 q^{76}-28 q^{74}+110 q^{72}-184 q^{70}+232 q^{68}-221 q^{66}+138 q^{64}+15 q^{62}-189 q^{60}+341 q^{58}-389 q^{56}+299 q^{54}-92 q^{52}-167 q^{50}+361 q^{48}-401 q^{46}+268 q^{44}-5 q^{42}-260 q^{40}+396 q^{38}-338 q^{36}+88 q^{34}+233 q^{32}-489 q^{30}+543 q^{28}-376 q^{26}+39 q^{24}+334 q^{22}-601 q^{20}+659 q^{18}-500 q^{16}+167 q^{14}+201 q^{12}-487 q^{10}+590 q^8-473 q^6+202 q^4+126 q^2-368+438 q^{-2} -298 q^{-4} +21 q^{-6} +276 q^{-8} -444 q^{-10} +409 q^{-12} -173 q^{-14} -149 q^{-16} +430 q^{-18} -533 q^{-20} +435 q^{-22} -184 q^{-24} -118 q^{-26} +340 q^{-28} -418 q^{-30} +346 q^{-32} -175 q^{-34} -7 q^{-36} +131 q^{-38} -178 q^{-40} +153 q^{-42} -91 q^{-44} +30 q^{-46} +14 q^{-48} -32 q^{-50} +28 q^{-52} -19 q^{-54} +9 q^{-56} -3 q^{-58} + q^{-60} }[/math]

Further Quantum Invariants

K11a158 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["K11a158"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ -t^4+5 t^3-14 t^2+25 t-29+25 t^{-1} -14 t^{-2} +5 t^{-3} - t^{-4} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ -z^8-3 z^6-4 z^4-2 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]=

{ 119, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ -q^4+4 q^3-8 q^2+13 q-16+19 q^{-1} -19 q^{-2} +16 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} +8 z^4+6 a^4 z^2-17 a^2 z^2-2 z^2 a^{-2} +11 z^2+3 a^4-7 a^2- a^{-2} +6 }[/math]

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\displaystyle{ 2 a^2 z^{10}+2 z^{10}+7 a^3 z^9+12 a z^9+5 z^9 a^{-1} +10 a^4 z^8+14 a^2 z^8+4 z^8 a^{-2} +8 z^8+9 a^5 z^7-7 a^3 z^7-30 a z^7-13 z^7 a^{-1} +z^7 a^{-3} +6 a^6 z^6-19 a^4 z^6-57 a^2 z^6-14 z^6 a^{-2} -46 z^6+3 a^7 z^5-14 a^5 z^5-12 a^3 z^5+10 a z^5+2 z^5 a^{-1} -3 z^5 a^{-3} +a^8 z^4-6 a^6 z^4+17 a^4 z^4+65 a^2 z^4+15 z^4 a^{-2} +56 z^4-2 a^7 z^3+12 a^5 z^3+18 a^3 z^3+10 a z^3+9 z^3 a^{-1} +3 z^3 a^{-3} -a^8 z^2+3 a^6 z^2-9 a^4 z^2-34 a^2 z^2-6 z^2 a^{-2} -27 z^2-4 a^5 z-7 a^3 z-5 a z-3 z a^{-1} -z a^{-3} +3 a^4+7 a^2+ a^{-2} +6 }[/math]

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a34,}

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["K11a158"];

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^3-14 t^2+25 t-29+25 t^{-1} -14 t^{-2} +5 t^{-3} - t^{-4} }[/math], [math]\displaystyle{ -q^4+4 q^3-8 q^2+13 q-16+19 q^{-1} -19 q^{-2} +16 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]=

{K11a34,}

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:

(-2, 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{ -8 }[/math] [math]\displaystyle{ 24 }[/math] [math]\displaystyle{ 32 }[/math] [math]\displaystyle{ \frac{164}{3} }[/math] [math]\displaystyle{ \frac{148}{3} }[/math] [math]\displaystyle{ -192 }[/math] [math]\displaystyle{ -432 }[/math] [math]\displaystyle{ -128 }[/math] [math]\displaystyle{ -136 }[/math] [math]\displaystyle{ -\frac{256}{3} }[/math] [math]\displaystyle{ 288 }[/math] [math]\displaystyle{ -\frac{1312}{3} }[/math] [math]\displaystyle{ -\frac{1184}{3} }[/math] [math]\displaystyle{ \frac{6449}{15} }[/math] [math]\displaystyle{ \frac{6884}{15} }[/math] [math]\displaystyle{ -\frac{25684}{45} }[/math] [math]\displaystyle{ \frac{2047}{9} }[/math] [math]\displaystyle{ -\frac{3631}{15} }[/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 K11a158. 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                     3   3 5                   5 1   -4 3                 8 3     5 1               8 5       -3 -1             11 8         3 -3           9 9           0 -5         7 10             -3 -7       5 9               4 -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}^{9}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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.

K11a157 K11a159