K11a73

K11a72

K11a74

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a73 at Knotilus!

Knot presentations

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

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

Three dimensional invariants

Symmetry type Reversible Unknotting number 1 3-genus 4 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a73/ThurstonBennequinNumber Hyperbolic Volume 18.4184 A-Polynomial See Data:K11a73/A-polynomial

[edit Notes for K11a73's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus [math]\displaystyle{ 4 }[/math] Rasmussen s-Invariant 0

[edit Notes for K11a73's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ t^4-7 t^3+21 t^2-37 t+45-37 t^{-1} +21 t^{-2} -7 t^{-3} + t^{-4} }[/math]
Conway polynomial	[math]\displaystyle{ z^8+z^6-z^4+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 177, 0 }
Jones polynomial	[math]\displaystyle{ -q^5+5 q^4-11 q^3+18 q^2-25 q+29-28 q^{-1} +25 q^{-2} -18 q^{-3} +11 q^{-4} -5 q^{-5} + q^{-6} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ z^8-2 a^2 z^6-z^6 a^{-2} +4 z^6+a^4 z^4-5 a^2 z^4-2 z^4 a^{-2} +5 z^4+a^4 z^2-a^2 z^2-a^4+3 a^2+ a^{-2} -2 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ 3 a^2 z^{10}+3 z^{10}+9 a^3 z^9+19 a z^9+10 z^9 a^{-1} +10 a^4 z^8+21 a^2 z^8+14 z^8 a^{-2} +25 z^8+5 a^5 z^7-8 a^3 z^7-23 a z^7+z^7 a^{-1} +11 z^7 a^{-3} +a^6 z^6-21 a^4 z^6-60 a^2 z^6-18 z^6 a^{-2} +5 z^6 a^{-4} -61 z^6-9 a^5 z^5-14 a^3 z^5-16 a z^5-26 z^5 a^{-1} -14 z^5 a^{-3} +z^5 a^{-5} -a^6 z^4+13 a^4 z^4+42 a^2 z^4+6 z^4 a^{-2} -4 z^4 a^{-4} +38 z^4+4 a^5 z^3+14 a^3 z^3+23 a z^3+18 z^3 a^{-1} +5 z^3 a^{-3} -2 a^4 z^2-5 a^2 z^2-3 z^2-2 a^3 z-4 a z-2 z a^{-1} -a^4-3 a^2- a^{-2} -2 }[/math]
The A2 invariant	[math]\displaystyle{ q^{18}-2 q^{16}-q^{14}+2 q^{12}-4 q^{10}+6 q^8+4 q^2-6+5 q^{-2} -5 q^{-4} + q^{-6} +3 q^{-8} -3 q^{-10} +3 q^{-12} - q^{-14} }[/math]
The G2 invariant	[math]\displaystyle{ q^{94}-4 q^{92}+11 q^{90}-24 q^{88}+36 q^{86}-43 q^{84}+30 q^{82}+20 q^{80}-101 q^{78}+212 q^{76}-304 q^{74}+312 q^{72}-190 q^{70}-98 q^{68}+491 q^{66}-854 q^{64}+1023 q^{62}-847 q^{60}+283 q^{58}+525 q^{56}-1305 q^{54}+1734 q^{52}-1579 q^{50}+834 q^{48}+250 q^{46}-1270 q^{44}+1787 q^{42}-1587 q^{40}+759 q^{38}+346 q^{36}-1218 q^{34}+1474 q^{32}-989 q^{30}-18 q^{28}+1115 q^{26}-1794 q^{24}+1733 q^{22}-901 q^{20}-408 q^{18}+1698 q^{16}-2465 q^{14}+2411 q^{12}-1509 q^{10}+86 q^8+1352 q^6-2298 q^4+2402 q^2-1665+400 q^{-2} +860 q^{-4} -1617 q^{-6} +1594 q^{-8} -856 q^{-10} -216 q^{-12} +1119 q^{-14} -1449 q^{-16} +1056 q^{-18} -149 q^{-20} -885 q^{-22} +1595 q^{-24} -1680 q^{-26} +1162 q^{-28} -252 q^{-30} -678 q^{-32} +1298 q^{-34} -1452 q^{-36} +1156 q^{-38} -583 q^{-40} -34 q^{-42} +503 q^{-44} -720 q^{-46} +692 q^{-48} -493 q^{-50} +241 q^{-52} -8 q^{-54} -149 q^{-56} +206 q^{-58} -199 q^{-60} +140 q^{-62} -72 q^{-64} +21 q^{-66} +16 q^{-68} -28 q^{-70} +27 q^{-72} -20 q^{-74} +10 q^{-76} -4 q^{-78} + q^{-80} }[/math]

Further Quantum Invariants

K11a73 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["K11a73"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ t^4-7 t^3+21 t^2-37 t+45-37 t^{-1} +21 t^{-2} -7 t^{-3} + t^{-4} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

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

{ 177, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ -q^5+5 q^4-11 q^3+18 q^2-25 q+29-28 q^{-1} +25 q^{-2} -18 q^{-3} +11 q^{-4} -5 q^{-5} + q^{-6} }[/math]

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

[math]\displaystyle{ z^8-2 a^2 z^6-z^6 a^{-2} +4 z^6+a^4 z^4-5 a^2 z^4-2 z^4 a^{-2} +5 z^4+a^4 z^2-a^2 z^2-a^4+3 a^2+ a^{-2} -2 }[/math]

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\displaystyle{ 3 a^2 z^{10}+3 z^{10}+9 a^3 z^9+19 a z^9+10 z^9 a^{-1} +10 a^4 z^8+21 a^2 z^8+14 z^8 a^{-2} +25 z^8+5 a^5 z^7-8 a^3 z^7-23 a z^7+z^7 a^{-1} +11 z^7 a^{-3} +a^6 z^6-21 a^4 z^6-60 a^2 z^6-18 z^6 a^{-2} +5 z^6 a^{-4} -61 z^6-9 a^5 z^5-14 a^3 z^5-16 a z^5-26 z^5 a^{-1} -14 z^5 a^{-3} +z^5 a^{-5} -a^6 z^4+13 a^4 z^4+42 a^2 z^4+6 z^4 a^{-2} -4 z^4 a^{-4} +38 z^4+4 a^5 z^3+14 a^3 z^3+23 a z^3+18 z^3 a^{-1} +5 z^3 a^{-3} -2 a^4 z^2-5 a^2 z^2-3 z^2-2 a^3 z-4 a z-2 z a^{-1} -a^4-3 a^2- a^{-2} -2 }[/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["K11a73"];

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^4-7 t^3+21 t^2-37 t+45-37 t^{-1} +21 t^{-2} -7 t^{-3} + t^{-4} }[/math], [math]\displaystyle{ -q^5+5 q^4-11 q^3+18 q^2-25 q+29-28 q^{-1} +25 q^{-2} -18 q^{-3} +11 q^{-4} -5 q^{-5} + q^{-6} }[/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:

(0, -1)

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{ 0 }[/math] [math]\displaystyle{ -8 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 16 }[/math] [math]\displaystyle{ 8 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ -\frac{80}{3} }[/math] [math]\displaystyle{ -\frac{32}{3} }[/math] [math]\displaystyle{ -8 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 32 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 88 }[/math] [math]\displaystyle{ -56 }[/math] [math]\displaystyle{ \frac{328}{3} }[/math] [math]\displaystyle{ -\frac{8}{3} }[/math] [math]\displaystyle{ 24 }[/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]0 is the signature of K11a73. 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 11                       1 -1 9                     4   4 7                   7 1   -6 5                 11 4     7 3               14 7       -7 1             15 11         4 -1           14 15           1 -3         11 14             -3 -5       7 14               7 -7     4 11                 -7 -9   1 7                   6 -11   4                     -4 -13 1                       1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-1 }[/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}^{4}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{14}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{14}\oplus{\mathbb Z}_2^{14} }[/math] [math]\displaystyle{ {\mathbb Z}^{14} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{15}\oplus{\mathbb Z}_2^{14} }[/math] [math]\displaystyle{ {\mathbb Z}^{15} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{14} }[/math] [math]\displaystyle{ {\mathbb Z}^{14} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/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.

K11a72 K11a74