K11a24

K11a23

K11a25

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a24 at Knotilus!

Knot presentations

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

A Braid Representative
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:K11a24/ThurstonBennequinNumber Hyperbolic Volume 17.9006 A-Polynomial See Data:K11a24/A-polynomial

[edit Notes for K11a24'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 K11a24's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ t^4-6 t^3+18 t^2-33 t+41-33 t^{-1} +18 t^{-2} -6 t^{-3} + t^{-4} }[/math]
Conway polynomial	[math]\displaystyle{ z^8+2 z^6+2 z^4+z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 157, 0 }
Jones polynomial	[math]\displaystyle{ q^6-4 q^5+9 q^4-16 q^3+22 q^2-25 q+26-22 q^{-1} +17 q^{-2} -10 q^{-3} +4 q^{-4} - q^{-5} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ z^8-a^2 z^6-2 z^6 a^{-2} +5 z^6-3 a^2 z^4-7 z^4 a^{-2} +z^4 a^{-4} +11 z^4-4 a^2 z^2-9 z^2 a^{-2} +2 z^2 a^{-4} +12 z^2-2 a^2-4 a^{-2} + a^{-4} +6 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ 2 z^{10} a^{-2} +2 z^{10}+8 a z^9+14 z^9 a^{-1} +6 z^9 a^{-3} +12 a^2 z^8+18 z^8 a^{-2} +7 z^8 a^{-4} +23 z^8+9 a^3 z^7+2 a z^7-12 z^7 a^{-1} -z^7 a^{-3} +4 z^7 a^{-5} +4 a^4 z^6-19 a^2 z^6-50 z^6 a^{-2} -14 z^6 a^{-4} +z^6 a^{-6} -58 z^6+a^5 z^5-13 a^3 z^5-26 a z^5-24 z^5 a^{-1} -21 z^5 a^{-3} -9 z^5 a^{-5} -4 a^4 z^4+15 a^2 z^4+42 z^4 a^{-2} +9 z^4 a^{-4} -2 z^4 a^{-6} +50 z^4-a^5 z^3+9 a^3 z^3+25 a z^3+30 z^3 a^{-1} +22 z^3 a^{-3} +7 z^3 a^{-5} +a^4 z^2-8 a^2 z^2-18 z^2 a^{-2} -3 z^2 a^{-4} +z^2 a^{-6} -23 z^2-3 a^3 z-8 a z-10 z a^{-1} -7 z a^{-3} -2 z a^{-5} +2 a^2+4 a^{-2} + a^{-4} +6 }[/math]
The A2 invariant	[math]\displaystyle{ -q^{14}+2 q^{12}-4 q^{10}+2 q^8+q^6-3 q^4+7 q^2-3+5 q^{-2} - q^{-4} -2 q^{-6} +3 q^{-8} -5 q^{-10} +2 q^{-12} - q^{-16} + q^{-18} }[/math]
The G2 invariant	[math]\displaystyle{ q^{80}-3 q^{78}+7 q^{76}-13 q^{74}+17 q^{72}-19 q^{70}+12 q^{68}+11 q^{66}-46 q^{64}+94 q^{62}-139 q^{60}+154 q^{58}-123 q^{56}+20 q^{54}+160 q^{52}-369 q^{50}+543 q^{48}-584 q^{46}+408 q^{44}-30 q^{42}-482 q^{40}+949 q^{38}-1167 q^{36}+1001 q^{34}-454 q^{32}-314 q^{30}+1007 q^{28}-1337 q^{26}+1155 q^{24}-520 q^{22}-297 q^{20}+930 q^{18}-1100 q^{16}+723 q^{14}+43 q^{12}-830 q^{10}+1288 q^8-1170 q^6+494 q^4+492 q^2-1389+1846 q^{-2} -1655 q^{-4} +877 q^{-6} +226 q^{-8} -1251 q^{-10} +1840 q^{-12} -1780 q^{-14} +1119 q^{-16} -121 q^{-18} -823 q^{-20} +1338 q^{-22} -1252 q^{-24} +639 q^{-26} +208 q^{-28} -899 q^{-30} +1117 q^{-32} -782 q^{-34} +40 q^{-36} +763 q^{-38} -1275 q^{-40} +1280 q^{-42} -790 q^{-44} +9 q^{-46} +733 q^{-48} -1173 q^{-50} +1190 q^{-52} -823 q^{-54} +270 q^{-56} +253 q^{-58} -591 q^{-60} +671 q^{-62} -539 q^{-64} +300 q^{-66} -52 q^{-68} -120 q^{-70} +194 q^{-72} -188 q^{-74} +134 q^{-76} -66 q^{-78} +16 q^{-80} +15 q^{-82} -26 q^{-84} +22 q^{-86} -16 q^{-88} +8 q^{-90} -3 q^{-92} + q^{-94} }[/math]

Further Quantum Invariants

K11a24 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["K11a24"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ t^4-6 t^3+18 t^2-33 t+41-33 t^{-1} +18 t^{-2} -6 t^{-3} + t^{-4} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ z^8+2 z^6+2 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]=

{ 157, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ q^6-4 q^5+9 q^4-16 q^3+22 q^2-25 q+26-22 q^{-1} +17 q^{-2} -10 q^{-3} +4 q^{-4} - q^{-5} }[/math]

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

[math]\displaystyle{ z^8-a^2 z^6-2 z^6 a^{-2} +5 z^6-3 a^2 z^4-7 z^4 a^{-2} +z^4 a^{-4} +11 z^4-4 a^2 z^2-9 z^2 a^{-2} +2 z^2 a^{-4} +12 z^2-2 a^2-4 a^{-2} + a^{-4} +6 }[/math]

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\displaystyle{ 2 z^{10} a^{-2} +2 z^{10}+8 a z^9+14 z^9 a^{-1} +6 z^9 a^{-3} +12 a^2 z^8+18 z^8 a^{-2} +7 z^8 a^{-4} +23 z^8+9 a^3 z^7+2 a z^7-12 z^7 a^{-1} -z^7 a^{-3} +4 z^7 a^{-5} +4 a^4 z^6-19 a^2 z^6-50 z^6 a^{-2} -14 z^6 a^{-4} +z^6 a^{-6} -58 z^6+a^5 z^5-13 a^3 z^5-26 a z^5-24 z^5 a^{-1} -21 z^5 a^{-3} -9 z^5 a^{-5} -4 a^4 z^4+15 a^2 z^4+42 z^4 a^{-2} +9 z^4 a^{-4} -2 z^4 a^{-6} +50 z^4-a^5 z^3+9 a^3 z^3+25 a z^3+30 z^3 a^{-1} +22 z^3 a^{-3} +7 z^3 a^{-5} +a^4 z^2-8 a^2 z^2-18 z^2 a^{-2} -3 z^2 a^{-4} +z^2 a^{-6} -23 z^2-3 a^3 z-8 a z-10 z a^{-1} -7 z a^{-3} -2 z a^{-5} +2 a^2+4 a^{-2} + a^{-4} +6 }[/math]

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a26, K11a315,}

Same Jones Polynomial (up to mirroring, [math]\displaystyle{ q\leftrightarrow q^{-1} }[/math]): {K11a26, K11a315,}

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["K11a24"];

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^4-6 t^3+18 t^2-33 t+41-33 t^{-1} +18 t^{-2} -6 t^{-3} + t^{-4} }[/math], [math]\displaystyle{ q^6-4 q^5+9 q^4-16 q^3+22 q^2-25 q+26-22 q^{-1} +17 q^{-2} -10 q^{-3} +4 q^{-4} - q^{-5} }[/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]=

{K11a26, K11a315,}

In[6]:=

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

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

Out[6]=

{K11a26, K11a315,}

Vassiliev invariants

V2 and V3:

(1, -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{ 4 }[/math] [math]\displaystyle{ -8 }[/math] [math]\displaystyle{ 8 }[/math] [math]\displaystyle{ -\frac{34}{3} }[/math] [math]\displaystyle{ -\frac{38}{3} }[/math] [math]\displaystyle{ -32 }[/math] [math]\displaystyle{ -\frac{80}{3} }[/math] [math]\displaystyle{ \frac{64}{3} }[/math] [math]\displaystyle{ -8 }[/math] [math]\displaystyle{ \frac{32}{3} }[/math] [math]\displaystyle{ 32 }[/math] [math]\displaystyle{ -\frac{136}{3} }[/math] [math]\displaystyle{ -\frac{152}{3} }[/math] [math]\displaystyle{ \frac{751}{30} }[/math] [math]\displaystyle{ -\frac{902}{15} }[/math] [math]\displaystyle{ \frac{3302}{45} }[/math] [math]\displaystyle{ -\frac{175}{18} }[/math] [math]\displaystyle{ \frac{271}{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]0 is the signature of K11a24. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -5 -4 -3 -2 -1 0 1 2 3 4 5 6 χ 13                       1 1 11                     3   -3 9                   6 1   5 7                 10 3     -7 5               12 6       6 3             13 10         -3 1           13 12           1 -1         10 14             4 -3       7 12               -5 -5     3 10                 7 -7   1 7                   -6 -9   3                     3 -11 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=-5 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{14}\oplus{\mathbb Z}_2^{12} }[/math] [math]\displaystyle{ {\mathbb Z}^{13} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{13} }[/math] [math]\displaystyle{ {\mathbb Z}^{13} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{12} }[/math] [math]\displaystyle{ {\mathbb Z}^{12} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=6 }[/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.

K11a23 K11a25