K11a25

K11a24

K11a26

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a25 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X8493 X12,5,13,6 X2837 X18,10,19,9 X20,11,21,12 X6,13,7,14 X10,16,11,15 X22,18,1,17 X14,19,15,20 X16,22,17,21
Gauss code	1, -4, 2, -1, 3, -7, 4, -2, 5, -8, 6, -3, 7, -10, 8, -11, 9, -5, 10, -6, 11, -9
Dowker-Thistlethwaite code	4 8 12 2 18 20 6 10 22 14 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:K11a25/ThurstonBennequinNumber Hyperbolic Volume 17.7863 A-Polynomial See Data:K11a25/A-polynomial

[edit Notes for K11a25'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 K11a25's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ -t^4+6 t^3-18 t^2+33 t-39+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	{ 155, 2 }
Jones polynomial	[math]\displaystyle{ -q^8+4 q^7-9 q^6+16 q^5-22 q^4+25 q^3-25 q^2+22 q-16+10 q^{-1} -4 q^{-2} + q^{-3} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ -z^8 a^{-2} -5 z^6 a^{-2} +2 z^6 a^{-4} +z^6-11 z^4 a^{-2} +7 z^4 a^{-4} -z^4 a^{-6} +3 z^4-12 z^2 a^{-2} +9 z^2 a^{-4} -2 z^2 a^{-6} +4 z^2-5 a^{-2} +4 a^{-4} - a^{-6} +3 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ 2 z^{10} a^{-2} +2 z^{10} a^{-4} +7 z^9 a^{-1} +14 z^9 a^{-3} +7 z^9 a^{-5} +19 z^8 a^{-2} +21 z^8 a^{-4} +10 z^8 a^{-6} +8 z^8+4 a z^7-6 z^7 a^{-1} -14 z^7 a^{-3} +4 z^7 a^{-5} +8 z^7 a^{-7} +a^2 z^6-55 z^6 a^{-2} -52 z^6 a^{-4} -12 z^6 a^{-6} +4 z^6 a^{-8} -18 z^6-8 a z^5-11 z^5 a^{-1} -17 z^5 a^{-3} -26 z^5 a^{-5} -11 z^5 a^{-7} +z^5 a^{-9} -2 a^2 z^4+50 z^4 a^{-2} +43 z^4 a^{-4} +5 z^4 a^{-6} -5 z^4 a^{-8} +15 z^4+5 a z^3+11 z^3 a^{-1} +21 z^3 a^{-3} +23 z^3 a^{-5} +7 z^3 a^{-7} -z^3 a^{-9} +a^2 z^2-24 z^2 a^{-2} -18 z^2 a^{-4} -z^2 a^{-6} +2 z^2 a^{-8} -8 z^2-a z-3 z a^{-1} -6 z a^{-3} -6 z a^{-5} -2 z a^{-7} +5 a^{-2} +4 a^{-4} + a^{-6} +3 }[/math]
The A2 invariant	[math]\displaystyle{ q^8-2 q^6+4 q^4-q^2+4 q^{-2} -6 q^{-4} +4 q^{-6} -4 q^{-8} + q^{-10} +2 q^{-12} -3 q^{-14} +5 q^{-16} -2 q^{-18} + q^{-22} - q^{-24} }[/math]
The G2 invariant	[math]\displaystyle{ q^{46}-3 q^{44}+8 q^{42}-16 q^{40}+23 q^{38}-28 q^{36}+20 q^{34}+11 q^{32}-66 q^{30}+145 q^{28}-216 q^{26}+228 q^{24}-143 q^{22}-63 q^{20}+357 q^{18}-625 q^{16}+753 q^{14}-615 q^{12}+190 q^{10}+409 q^8-975 q^6+1270 q^4-1120 q^2+547+253 q^{-2} -963 q^{-4} +1286 q^{-6} -1080 q^{-8} +449 q^{-10} +337 q^{-12} -920 q^{-14} +1032 q^{-16} -632 q^{-18} -121 q^{-20} +884 q^{-22} -1316 q^{-24} +1205 q^{-26} -568 q^{-28} -380 q^{-30} +1275 q^{-32} -1781 q^{-34} +1685 q^{-36} -1007 q^{-38} -25 q^{-40} +1032 q^{-42} -1653 q^{-44} +1665 q^{-46} -1082 q^{-48} +174 q^{-50} +693 q^{-52} -1162 q^{-54} +1067 q^{-56} -488 q^{-58} -276 q^{-60} +875 q^{-62} -1027 q^{-64} +679 q^{-66} +6 q^{-68} -720 q^{-70} +1166 q^{-72} -1164 q^{-74} +750 q^{-76} -106 q^{-78} -527 q^{-80} +917 q^{-82} -982 q^{-84} +755 q^{-86} -353 q^{-88} -53 q^{-90} +345 q^{-92} -475 q^{-94} +442 q^{-96} -312 q^{-98} +147 q^{-100} -94 q^{-104} +128 q^{-106} -121 q^{-108} +88 q^{-110} -46 q^{-112} +15 q^{-114} +8 q^{-116} -17 q^{-118} +16 q^{-120} -13 q^{-122} +7 q^{-124} -3 q^{-126} + q^{-128} }[/math]

Further Quantum Invariants

K11a25 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["K11a25"];

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-39+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]=

{ 155, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ -q^8+4 q^7-9 q^6+16 q^5-22 q^4+25 q^3-25 q^2+22 q-16+10 q^{-1} -4 q^{-2} + q^{-3} }[/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} -5 z^6 a^{-2} +2 z^6 a^{-4} +z^6-11 z^4 a^{-2} +7 z^4 a^{-4} -z^4 a^{-6} +3 z^4-12 z^2 a^{-2} +9 z^2 a^{-4} -2 z^2 a^{-6} +4 z^2-5 a^{-2} +4 a^{-4} - a^{-6} +3 }[/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} a^{-4} +7 z^9 a^{-1} +14 z^9 a^{-3} +7 z^9 a^{-5} +19 z^8 a^{-2} +21 z^8 a^{-4} +10 z^8 a^{-6} +8 z^8+4 a z^7-6 z^7 a^{-1} -14 z^7 a^{-3} +4 z^7 a^{-5} +8 z^7 a^{-7} +a^2 z^6-55 z^6 a^{-2} -52 z^6 a^{-4} -12 z^6 a^{-6} +4 z^6 a^{-8} -18 z^6-8 a z^5-11 z^5 a^{-1} -17 z^5 a^{-3} -26 z^5 a^{-5} -11 z^5 a^{-7} +z^5 a^{-9} -2 a^2 z^4+50 z^4 a^{-2} +43 z^4 a^{-4} +5 z^4 a^{-6} -5 z^4 a^{-8} +15 z^4+5 a z^3+11 z^3 a^{-1} +21 z^3 a^{-3} +23 z^3 a^{-5} +7 z^3 a^{-7} -z^3 a^{-9} +a^2 z^2-24 z^2 a^{-2} -18 z^2 a^{-4} -z^2 a^{-6} +2 z^2 a^{-8} -8 z^2-a z-3 z a^{-1} -6 z a^{-3} -6 z a^{-5} -2 z a^{-7} +5 a^{-2} +4 a^{-4} + a^{-6} +3 }[/math]

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a19, K11a281,}

Same Jones Polynomial (up to mirroring, [math]\displaystyle{ q\leftrightarrow q^{-1} }[/math]): {K11a19,}

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["K11a25"];

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-39+33 t^{-1} -18 t^{-2} +6 t^{-3} - t^{-4} }[/math], [math]\displaystyle{ -q^8+4 q^7-9 q^6+16 q^5-22 q^4+25 q^3-25 q^2+22 q-16+10 q^{-1} -4 q^{-2} + q^{-3} }[/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]=

{K11a19, K11a281,}

In[6]:=

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

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

Out[6]=

{K11a19,}

Vassiliev invariants

V2 and V3:

(-1, 0)

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{ 0 }[/math] [math]\displaystyle{ 8 }[/math] [math]\displaystyle{ \frac{130}{3} }[/math] [math]\displaystyle{ \frac{62}{3} }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 32 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ -\frac{32}{3} }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ -\frac{520}{3} }[/math] [math]\displaystyle{ -\frac{248}{3} }[/math] [math]\displaystyle{ -\frac{7471}{30} }[/math] [math]\displaystyle{ \frac{1822}{15} }[/math] [math]\displaystyle{ -\frac{17822}{45} }[/math] [math]\displaystyle{ \frac{1135}{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 K11a25. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -4 -3 -2 -1 0 1 2 3 4 5 6 7 χ 17                       1 -1 15                     3   3 13                   6 1   -5 11                 10 3     7 9               12 6       -6 7             13 10         3 5           12 12           0 3         10 13             -3 1       7 13               6 -1     3 9                 -6 -3   1 7                   6 -5   3                     -3 -7 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=3 }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{12} }[/math] [math]\displaystyle{ {\mathbb Z}^{12} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{13} }[/math] [math]\displaystyle{ {\mathbb Z}^{13} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{12} }[/math] [math]\displaystyle{ {\mathbb Z}^{12} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=7 }[/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.

K11a24 K11a26