K11a125

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K11a124.gif

K11a124

K11a126.gif

K11a126

K11a125.gif
(Knotscape image) 		See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a125 at Knotilus!

[edit K11a125 Quick Notes]


[edit K11a125 Further Notes and Views]


Knot presentations

Planar diagram presentation X4251 X10,4,11,3 X14,5,15,6 X20,8,21,7 X2,10,3,9 X16,11,17,12 X18,14,19,13 X8,15,9,16 X22,17,1,18 X6,20,7,19 X12,22,13,21
Gauss code 1, -5, 2, -1, 3, -10, 4, -8, 5, -2, 6, -11, 7, -3, 8, -6, 9, -7, 10, -4, 11, -9
Dowker-Thistlethwaite code 4 10 14 20 2 16 18 8 22 6 12
A Braid Representative
BraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart3.gif
BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart1.gifBraidPart4.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart2.gifBraidPart0.gif
A Morse Link Presentation K11a125 ML.gif

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:K11a125/ThurstonBennequinNumber
Hyperbolic Volume 18.6455
A-Polynomial See Data:K11a125/A-polynomial

[edit Notes for K11a125'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 K11a125's four dimensional invariants]

Polynomial invariants

Alexander polynomial [math]\displaystyle{ -t^4+6 t^3-19 t^2+38 t-47+38 t^{-1} -19 t^{-2} +6 t^{-3} - t^{-4} }[/math]
Conway polynomial [math]\displaystyle{ -z^8-2 z^6-3 z^4+1 }[/math]
2nd Alexander ideal (db, data sources) [math]\displaystyle{ \{1\} }[/math]
Determinant and Signature { 175, 2 }
Jones polynomial [math]\displaystyle{ -q^8+5 q^7-12 q^6+19 q^5-25 q^4+29 q^3-28 q^2+24 q-17+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} +6 z^4 a^{-4} -z^4 a^{-6} +3 z^4-10 z^2 a^{-2} +7 z^2 a^{-4} -z^2 a^{-6} +4 z^2-3 a^{-2} +3 a^{-4} - a^{-6} +2 }[/math]
Kauffman polynomial (db, data sources) [math]\displaystyle{ 3 z^{10} a^{-2} +3 z^{10} a^{-4} +8 z^9 a^{-1} +19 z^9 a^{-3} +11 z^9 a^{-5} +18 z^8 a^{-2} +26 z^8 a^{-4} +16 z^8 a^{-6} +8 z^8+4 a z^7-9 z^7 a^{-1} -27 z^7 a^{-3} -2 z^7 a^{-5} +12 z^7 a^{-7} +a^2 z^6-55 z^6 a^{-2} -66 z^6 a^{-4} -24 z^6 a^{-6} +5 z^6 a^{-8} -17 z^6-8 a z^5-5 z^5 a^{-1} -3 z^5 a^{-3} -22 z^5 a^{-5} -15 z^5 a^{-7} +z^5 a^{-9} -2 a^2 z^4+50 z^4 a^{-2} +49 z^4 a^{-4} +12 z^4 a^{-6} -3 z^4 a^{-8} +14 z^4+5 a z^3+8 z^3 a^{-1} +12 z^3 a^{-3} +14 z^3 a^{-5} +5 z^3 a^{-7} +a^2 z^2-20 z^2 a^{-2} -16 z^2 a^{-4} -4 z^2 a^{-6} -7 z^2-a z-2 z a^{-1} -2 z a^{-3} -2 z a^{-5} -z a^{-7} +3 a^{-2} +3 a^{-4} + a^{-6} +2 }[/math]
The A2 invariant [math]\displaystyle{ q^8-2 q^6+4 q^4-2 q^2-1+5 q^{-2} -6 q^{-4} +5 q^{-6} -3 q^{-8} + q^{-10} +3 q^{-12} -4 q^{-14} +5 q^{-16} -3 q^{-18} - q^{-20} +2 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}+10 q^{32}-62 q^{30}+137 q^{28}-208 q^{26}+233 q^{24}-176 q^{22}-2 q^{20}+294 q^{18}-608 q^{16}+833 q^{14}-812 q^{12}+451 q^{10}+199 q^8-958 q^6+1532 q^4-1632 q^2+1151-188 q^{-2} -916 q^{-4} +1717 q^{-6} -1870 q^{-8} +1282 q^{-10} -198 q^{-12} -912 q^{-14} +1540 q^{-16} -1415 q^{-18} +595 q^{-20} +565 q^{-22} -1513 q^{-24} +1825 q^{-26} -1319 q^{-28} +142 q^{-30} +1226 q^{-32} -2263 q^{-34} +2534 q^{-36} -1901 q^{-38} +590 q^{-40} +955 q^{-42} -2166 q^{-44} +2616 q^{-46} -2157 q^{-48} +971 q^{-50} +450 q^{-52} -1558 q^{-54} +1919 q^{-56} -1425 q^{-58} +370 q^{-60} +777 q^{-62} -1465 q^{-64} +1405 q^{-66} -655 q^{-68} -462 q^{-70} +1422 q^{-72} -1815 q^{-74} +1496 q^{-76} -606 q^{-78} -462 q^{-80} +1302 q^{-82} -1636 q^{-84} +1413 q^{-86} -801 q^{-88} +61 q^{-90} +535 q^{-92} -847 q^{-94} +837 q^{-96} -597 q^{-98} +284 q^{-100} +7 q^{-102} -189 q^{-104} +249 q^{-106} -230 q^{-108} +153 q^{-110} -72 q^{-112} +14 q^{-114} +22 q^{-116} -31 q^{-118} +28 q^{-120} -20 q^{-122} +10 q^{-124} -4 q^{-126} + q^{-128} }[/math]
Further Quantum Invariants
K11a125 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["K11a125"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= [math]\displaystyle{ -t^4+6 t^3-19 t^2+38 t-47+38 t^{-1} -19 t^{-2} +6 t^{-3} - t^{-4} }[/math]
In[5]:= Conway[K][z]
Out[5]= [math]\displaystyle{ -z^8-2 z^6-3 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]= { 175, 2 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= [math]\displaystyle{ -q^8+5 q^7-12 q^6+19 q^5-25 q^4+29 q^3-28 q^2+24 q-17+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} +6 z^4 a^{-4} -z^4 a^{-6} +3 z^4-10 z^2 a^{-2} +7 z^2 a^{-4} -z^2 a^{-6} +4 z^2-3 a^{-2} +3 a^{-4} - a^{-6} +2 }[/math]
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= [math]\displaystyle{ 3 z^{10} a^{-2} +3 z^{10} a^{-4} +8 z^9 a^{-1} +19 z^9 a^{-3} +11 z^9 a^{-5} +18 z^8 a^{-2} +26 z^8 a^{-4} +16 z^8 a^{-6} +8 z^8+4 a z^7-9 z^7 a^{-1} -27 z^7 a^{-3} -2 z^7 a^{-5} +12 z^7 a^{-7} +a^2 z^6-55 z^6 a^{-2} -66 z^6 a^{-4} -24 z^6 a^{-6} +5 z^6 a^{-8} -17 z^6-8 a z^5-5 z^5 a^{-1} -3 z^5 a^{-3} -22 z^5 a^{-5} -15 z^5 a^{-7} +z^5 a^{-9} -2 a^2 z^4+50 z^4 a^{-2} +49 z^4 a^{-4} +12 z^4 a^{-6} -3 z^4 a^{-8} +14 z^4+5 a z^3+8 z^3 a^{-1} +12 z^3 a^{-3} +14 z^3 a^{-5} +5 z^3 a^{-7} +a^2 z^2-20 z^2 a^{-2} -16 z^2 a^{-4} -4 z^2 a^{-6} -7 z^2-a z-2 z a^{-1} -2 z a^{-3} -2 z a^{-5} -z a^{-7} +3 a^{-2} +3 a^{-4} + a^{-6} +2 }[/math]

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, [math]\displaystyle{ q\leftrightarrow q^{-1} }[/math]): {K11a297,}

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["K11a125"];
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^3-19 t^2+38 t-47+38 t^{-1} -19 t^{-2} +6 t^{-3} - t^{-4} }[/math], [math]\displaystyle{ -q^8+5 q^7-12 q^6+19 q^5-25 q^4+29 q^3-28 q^2+24 q-17+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]= {}
In[6]:= DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]
KnotTheory::loading: Loading precomputed data in Jones4Knots11`.
Out[6]= {K11a297,}

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{ 48 }[/math] [math]\displaystyle{ 24 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ \frac{272}{3} }[/math] [math]\displaystyle{ \frac{32}{3} }[/math] [math]\displaystyle{ 40 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 32 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 168 }[/math] [math]\displaystyle{ 72 }[/math] [math]\displaystyle{ -24 }[/math] [math]\displaystyle{ 40 }[/math] [math]\displaystyle{ -40 }[/math]

V2,1 through V6,9 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 K11a125. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -4-3-2-101234567χ
17           1-1
15          4 4
13         81 -7
11        114  7
9       148   -6
7      1511    4
5     1314     1
3    1115      -4
1   714       7
-1  310        -7
-3 17         6
-5 3          -3
-71           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}^{10}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math]
[math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{14}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math]
[math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{15}\oplus{\mathbb Z}_2^{13} }[/math] [math]\displaystyle{ {\mathbb Z}^{13} }[/math]
[math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{14}\oplus{\mathbb Z}_2^{15} }[/math] [math]\displaystyle{ {\mathbb Z}^{15} }[/math]
[math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{14} }[/math] [math]\displaystyle{ {\mathbb Z}^{14} }[/math]
[math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math]
[math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math]
[math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/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.

K11a124.gif

K11a124

K11a126.gif

K11a126

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